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# Annotated version of "What's Special About This Number?" (Part 0)

## Introduction

Erich Friedman has a very nice (and deservedly popular) page called
**What's Special About This Number?**

It does not, however, mention the sequences in the OEIS where these numbers can be found (and from where I suspect most of the entries were taken).

The present set of ten pages is a snapshot of his page as of May 23, 2010, with pointers to the corresponding entries in the OEIS. The pages are:

- Part 0: 0 to 999,
- Part 1: 1000 to 1999,
- Part 2: 2000 to 2999,
- Part 3: 3000 to 3999,
- Part 4: 4000 to 4999,
- Part 5: 5000 to 5999,
- Part 6: 6000 to 6999,
- Part 7: 7000 to 7999,
- Part 8: 8000 to 8999,
- Part 9: 9000 to 9999.

People are invited to add more pointers to these pages, by adding the appropriate A-numbers to the entries. It may be necessary to create new sequences to do this - see A158304 for an example.

To add a link to sequence A000108, for example, type A000108.

I should add that this is being done with Erich Friedman's approval.

I did not do a very good job of converting the original html format to wiki format, and in some cases you may have to refer to Erich's page to figure out the meaning or the links.

You may well find better descriptions for some numbers. If so, please send them to Erich and make the corresponding changes here. (The wiki software is complaining that these pages are too long. I decided to ignore these complaints.)

Neil Sloane

## Part 0: The Numbers 0 to 999

**0** is the additive identity and the empty sum (A000004)

**1** is the multiplicative identity and the empty product (A000012)

**2** is the only even prime (A000040). It is also the only positive integer which satisfies x + x = x ∗ x = x ^ x = x ↑↑ x = x ↑↑↑ x = ... (see Knuth's up-arrow notation). It is also the 1^{st} primary pseudoperfect number (A054377).

**3** is the number of spatial dimensions we live in. It is also the number of regular polytopes in all dimensions larger than four (A060296)

**4** is the smallest number of colors sufficient to color all planar maps (A000934, A158411)

**5** is the number of Platonic solids, i.e., the number of regular polytopes in 3 dimensions (A060296(3)=5)

**6** is the smallest perfect number (A000396). Also, 6 is the maximum (finite) value of A060296,
as there are six regular polytopes in 4 dimensions.
Among symmetric groups, only has a non-trivial outer automorphism. It is also the 2^{nd} primary pseudoperfect number (A054377).

**7** is the smallest number of sides of a regular polygon that is not constructable by straight-edge and compass (A003401, A004169)

**8** is the largest cube in the Fibonacci sequence (A000045, A000578)

**9** is the maximum number of cubes that are needed to sum to any positive integer (A002376)

**10** is the base of our number system

**11** is the largest known multiplicative persistence (A003001, A031346)

**12** is the smallest abundant number (A005101)

**13** is the number of Archimedian solids

**14** is the smallest number n with the property that there are no numbers relatively prime to n smaller numbers (A005277)

**15** is the smallest composite number n with the property that there is only one group of order n (A000001, A050384)

**16** is the only number of the form x^{y} = y^{x} with x and y different integers

**17** is the number of wallpaper groups

**18** is the only number (other than 0) that is twice the sum of its digits (A169805, A007953)

**19** is the maximum number of 4^{th} powers needed to sum to any number (A002377)

**20** is the number of rooted trees with 6 vertices (A000081)

**21** is the smallest number of distinct squares needed to tile a square (A006983)

**22** is the number of partitions of 8 (A000041)

**23** is the smallest number of integer -sided boxes that tile a box so that no two boxes share a common length

**24** is the largest number divisible by all numbers less than its square root

**25** is the smallest square that can be written as a sum of 2 (nonzero) squares (A134422).
It is also the smallest composite number that is both a
Lucky number (A000959)
and Ludic number (A003309). Also, the smallest composite number,
whose binary expansion 11001, when interpreted as a polynomial in
ring GF(2)[X], (x^4 + x^3 + 1), is irreducible in that ring (A091214).

**26** is the only positive number to be directly between a square and a cube

**27** is the largest number that is the sum of the digits of its cube

**28** is the 2^{nd} perfect number (A000396)

**29** is the 7^{th} Lucas number (A000032)

**30** is the largest number with the property that all smaller numbers relatively prime to it are prime

**31** is a Mersenne prime (A000043, A000668)

**32** is the smallest non-trivial 5^{th} power (A000584)

**33** is the largest number that is not a sum of distinct triangular numbers

**34** is the smallest number with the property that it and its neighbors have the same number of divisors (A169834, A005238)

**35** is the number of hexominoes (A000105)

**36** is the smallest non-trivial number which is both square (A000290) and triangular (A000217) (I.e. is in A001110)

**37** is the maximum number of 5^{th} powers needed to sum to any number. It is also the third (and probably the last) prime of the form 4k+1, for which all sums Sum_{j=1..u} L(j/p) (with u ranging from 1 to (p-1)/2) are nonnegative, where L(j/p) is Legendre symbol of j and p. (A080114)

**38** is the last Roman numeral when written lexicographically

**39** is the smallest number which has 3 different partitions into 3 parts with the same product

**40** is the only number whose letters are in alphabetical order

**41** is a value of n so that x^{2} + x + n takes on prime values for x = 0, 1, 2, ... n-2

**42** is the 5^{th} Catalan number (A000108), and the last one which is an integral multiple of the immediately preceding Catalan number (C_{4} = 14). It is also the 3^{rd} primary pseudoperfect number (A054377).

**43** is the number of sided 7-iamonds

**44** is the number of derangements of 5 items (A000166)

**45** is a Kaprekar number (A006886)

**46** is the number of different arrangements (up to rotation and reflection) of 9 non-attacking queens on a 9×9 chessboard

**47** is the largest number of cubes that cannot tile a cube

**48** is the smallest number with 10 divisors

**49** is the smallest number with the property that it and its neighbors are squareful

**50** is the smallest number that can be written as the sum of of 2 squares in 2 ways

**51** is the 6^{th} Motzkin number

**52** is the 5^{th} Bell number (A000110)

**53** is the only two digit number that is reversed in hexadecimal

**54** is the smallest number that can be written as the sum of 3 squares in 3 ways

**55** is the largest triangular number (A000217) in the Fibonacci sequence (A000045). It is the tenth term (after the zero) in both sequences, 55 = A000217(10) = A000045(10)

**56** is the number of reduced 5×5 Latin squares A000315)

**57** = 111 in base 7 (A023000)

**58** is the number of commutative semigroups of order 4

**59** is the number of stellations of an icosahedron

**60** is the smallest number divisible by 1 through 6 (A169823).

**60** is also the order of the smallest non-cyclic simple group (A001034)

**61** is the 3^{rd} secant number (A000364)

**62** is the smallest number that can be written as the sum of of 3 distinct squares in 2 ways

**63** is the number of partially ordered sets on 5 elements (A000112)

**64** is the smallest number with 7 divisors

**65** is the smallest number that becomes square if its reverse is either added to or subtracted from it

**66** is the number of 8-iamonds

**67** is the smallest number which is palindromic in bases 5 and 6

**68** is the 2-digit string that appears latest in the decimal expansion of π

**69** has the property that n^{2} and n^{3} together contain each digit once

**70** is the smallest weird number

**71** divides the sum of the primes less than it

**72** is the maximum number of spheres that can touch another sphere in a lattice packing in 6 dimensions (A001116)

**73** is the smallest multi-digit number which is one less than twice its reverse (A169830)

**74** is the number of different non-Hamiltonian polyhedra with a minimum number of vertices

**75** is the number of orderings of 4 objects with ties allowed (A000670)

**76** is an automorphic number

**77** is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1

**78** is the smallest number that can be written as the sum of of 4 distinct squares in 3 ways

**79** is a permutable prime

**80** is the smallest number n where n and n+1 are both products of 4 or more primes

**81** is the square of the sum of its digits

**82** is the number of 6-hexes

**83** is the number of strongly connected digraphs with 4 vertices

**84** is the largest order of a permutation of 14 elements

**85** is the largest n for which 1^{2}+2^{2}+3^{2}+ ... +n^{2} = 1+2+3+ ... +m has a solution

**86** = 222 in base 6

**87** is the sum of the squares of the first 4 primes

**88** is the only number known whose square has no isolated digits

**89** = 8^{1} + 9^{2} (A028387)

**90** is the number of degrees in a right angle

**91** is the smallest pseudoprime in base 3

**92** is the number of different arrangements of 8 non-attacking queens on an 8×8 chessboard

**93** = 333 in base 5

**94** is a Smith number

**95** is the number of planar partitions of 10

**96** is the smallest number that can be written as the difference of 2 squares in 4 ways

**97** is the smallest number with the property that its first 3 multiples contain the digit 9

**98** is the smallest number with the property that its first 5 multiples contain the digit 9

**99** is a Kaprekar number (A006886)

**100** is the smallest square which is also the sum of 4 consecutive non-zero cubes A027603 (Are there others?)

**101** is the number of partitions of 13

**102** is the smallest number with three different digits

**103** has the property that placing the last digit first gives 1 more than triple it

**104** is the smallest known number of unit line segments that can exist in the plane, 4 touching at every vertex

**105** is the largest number n known with the property that n - 2^{k} is prime for k>1.
Also, as it is the first natural number to have three distinct odd prime factors (105 = 3*5*7), the 105th cyclotomic polynomial () is the first one which has a
coefficient with absolute value larger than 1 (A013594).

**106** is the number of trees with 10 vertices

**107** is the exponent of a Mersenne prime (A000043, A000668)

**108** is 3 hyperfactorial

**108** is also the number of heptominoes (A000105)

**109** has a 5^{th} root that starts 2.555555...

**110** is the smallest number that is the product of two different substrings

**111** is the smallest possible magic constant of a 3×3 magic square of distinct primes

**112** is the side of the smallest square that can be tiled with distinct integer -sided squares

**113** is a permutable prime

**114** = 222 in base 7 (A125725)

**115** is the number of rooted trees with 8 vertices (A000081)

**116** is a value of n for which n! + 1 is prime

**117** is the smallest possible value of the longest edge in a Heronian Tetrahedron

**118** is the smallest number that has 4 different partitions into 3 parts with the same product

**119** is the smallest number n where either n or n+1 is divisible by the numbers from 1 to 8

**120** is the smallest number to appear 6 times in Pascal's triangle. It is also the third (and probably the last?) number which is both factorial (A000142) and triangular number (A000217)

**121** is the only square known of the form 1 + p + p^{2} + p^{3} + p^{4}, where p is prime

**122** is the smallest number n>1 so that n concatenated with n-1 0's concatenated with the reverse of n is prime

**123** is the 10^{th} Lucas number (A000032)

**124** is the smallest number with the property that its first 3 multiples contain the digit 2

**125** is the only number known that contains all its proper divisors as proper substrings

**126** = [1] binomial(9,4) (A007318)

**127** is a Mersenne prime (A000043, A000668)

**128** is the largest number which is not the sum of distinct squares

**129** is the smallest number that can be written as the sum of 3 squares in 4 ways

**130** is the number of functions from 6 unlabeled points to themselves

**131** is a permutable prime

**132** is the smallest number which is the sum of all of the 2-digit numbers that can be formed with its digits

**133** is the smallest number n for which the sum of the proper divisors of n divides φ (n)

**134** = _{8}C _{1} + _{8}C _{3} + _{8}C _{4}

**135** = 1^{1} + 3^{2} + 5^{3}

**136** is the sum of the cubes of the digits of the sum of the cubes of its digits

**137** is the smallest prime with 3 distinct digits that remains prime if one of its digits is removed

**138** is a value of n for which n!!! - 1 is prime

**139** is the number of unlabeled topologies with 5 elements

**140** is a harmonic divisor number

**141** is the 6^{th} central trinomial coefficient

**142** is the number of planar graphs with 6 vertices

**143** is the smallest quasi-Carmichael number in base 8

**144** is the largest square (A000290) in the Fibonacci sequence (A000045). It is the twelfth term (after the zero) in both sequences, 144 = A000290(12) = A000045(12)

**145** is a factorion

**146** = 222 in base 8

**147** is the number of sided 6-hexes

**148** is the number of perfect graphs with 6 vertices

**149** is the smallest number whose square begins with three 2's

**150** = 10010110_{2} = 2112_{4} = 1100_{5}, each using 2 different digits an equal number of times

**151** is a palindromic prime (A002385)

**152** has a square composed of the digits 0-4 (A158304)

**153** is a narcissistic number (A005188)

**154** is the smallest number which is palindromic in bases 6, 8, and 9

**155** is the sum of the primes between its smallest and largest prime factor (A169802, A055233)

**156** is the number of graphs with 6 vertices (A000088)

**157** is a number whose square contains the same digits as the square of its successor (A072841 (the defintion was wrong)

**158** is the number of planar partitions of 11 (A000219)

**159** is the number of isomers of C_{11}H_{24}

**160** is the number of 9-iamonds

**161** is a Cullen number

**162** is the smallest number that can be written as the sum of of 4 positive squares in 9 ways

**163** is the largest Heegner Number

**164** is the smallest number which is the concatenation of squares in two different ways

**165** is the midpoint of the n^{th} larger prime and n^{th} smaller prime, for 1 ≤ n ≤ 6

**166** is the number of monotone Boolean functions of 4 variables

**167** is the smallest number whose 4^{th} power begins with 4 identical digits

**168** is the size of the smallest non-cyclic simple group which is not an alternating group (A001034)

**169** is the 7^{th} Pell number. Apparently, the largest square (13x13) in A001653.

**170** is the smallest number n for which φ (n) and σ (n) are both square

**171** has the same number of digits in Roman numerals as its cube

**172** = 444 in base 6

**173** has a square containing only 2 digits

**174** is the smallest number that can be written as the sum of of 4 positive distinct squares in 6 ways

**175** = 1^{1} + 7^{2} + 5^{3}

**176** is an octagonal pentagonal number

**177** is the number of graphs with 7 edges

**178** has a cube with the same digits as another cube

**179** has a square comprised of the digits 0-4

**180** is the total number of degrees in a triangle

**181** is a strobogrammatic prime

**182** is the number of connected bipartite graphs with 8 vertices

**183** is the smallest number n so that n concatenated with n+1 is square

**184** is a Kaprekar constant in base 3

**185** is the number of conjugacy classes in the automorphism group of the 8 dimensional hypercube .

**186** is the number of degree 11 irreducible polynomials over GF(2)

**187** is the smallest quasi-Carmichael number in base 7

**188** is the number of semigroups of order 4

**189** is a Kaprekar constant in base 2

**190** is the largest number with the property that it and its distinct prime factors are palindromic in Roman numerals

**191** is a number n for which n, n+2, n+6, and n+8 are all prime

**192** is the smallest number with 14 divisors

**193** is the largest number that can be written as ab + ac + bc with 0 < a < b < c in a unique way

**194** is the smallest number that can be written as the sum of 3 squares in 5 ways

**195** is the smallest value of n such that _{2n}C _{n} is divisible by n^{2}

**196** is the smallest number that is not known to reach a palindrome when repeatedly added to its reverse

**197** is a Keith number

**198** = 11 + 99 + 88

**199** is the 11^{th} Lucas number (A000032)

**200** is the smallest number which can not be made prime by changing one of its digits

**201** is a Kaprekar constant in base 4

**202** has a cube that contains only even digits

**203** is the 6^{th} Bell number (A000110)

**204** is the square root of a triangular number

**205** = 5 × 41 = 541_{6}

**206** is the smallest number whose English name contains all five vowels exactly once

**207** has a 4^{th} power where the first half of the digits are a permutation of the last half of the digits

**208** is the 10^{th} Tetranacci number

**209** is the smallest quasi-Carmichael number in base 9

**210** is the product of the first 4 primes

**211** has a cube containing only 3 different digits

**212** has a square with 4/5 of the digits are the same

**213** is the number of perfect squared rectangles of order 13

**214** is a value of n for which n!! - 1 is prime

**215** = 555 in base 6

**216** is the smallest cube that can be written as the sum of 3 cubes

**217** is a Kaprekar constant in base 2

**218** is the number of digraphs with 4 vertices

**219** is the number of space groups , not including handedness

**220** is the smallest amicable number

**221** is the number of Hamiltonian planar graphs with 7 vertices

**222** is the number of lattices on 8 unlabeled nodes

**223** is the smallest prime p which has more primitive roots below p/2 than above p/2

**224** is the Entringer number E(6,3).

**225** is an octagonal square number

**226** are the first 3 digits of π ^{226}

**227** is the number of connected planar graphs with 8 edges

**228** is the number of ways, up to rotation and reflection, of dissecting a regular 11-gon into 9 triangles

**229** is the smallest prime that remains prime when added to its reverse

**230** is the number of space groups , including handedness

**231** is the number of partitions of 16

**232** is the number of 7×7 symmetric permutation matrices

**233** is the smallest number with the property that it and its neighbors can be written as a sum of 2 squares

**234** is the number of ways to stack 12 pennies in a line so that each penny lies on the table or on two pennies

**235** is the number of trees with 11 vertices

**236** is the number of possible positions in Othello after 2 moves by both players

**237** is the smallest number with the property that its first 3 multiples contain the digit 7

**238** is the number of connected partial orders on 6 unlabeled elements

**239** is the largest number that cannot be written as a sum of 8 or fewer cubes .

**240** is the smallest number with 20 divisors

**241** is the only number n for which the n^{th} prime is π(n π(n))

**242** is the smallest n for which n, n+1, n+2, and n+3 have the same number of divisors

**243** = 3^{5} (A000032)

**244** is the smallest number (besides 2) that can be written as the sum of 2 squares or the sum of two 5^{th} powers

**245** is a stella octangula number

**246** = _{9}C _{2} + _{9}C _{4} + _{9}C _{6}

**247** is the smallest possible difference between two integers that together contain each digit exactly once

**248** is the smallest number n>1 for which the arithmetic , geometric , and harmonic means of φ (n) and σ (n) are all integers

**249** is the index of a prime Woodall number

**250** is the smallest multi-digit number so that the sum of the squares of its prime factors equals the sum of the squares of its digits

**251** is the smallest number that can be written as the sum of 3 cubes in 2 ways

**252** is the 5^{th} central binomial coefficient

**253** is the smallest non-trivial triangular star number

**254** is the smallest multi-digit composite number all of whose proper divisors contain the digit 2

**255** = 11111111 in base 2 (A000225)

**256** is the smallest non-trivial 8^{th} power

**257** is a Fermat prime

**258** is a value of n so that n(n+9) is a palindrome

**259** = 1111 in base 6

**260** is the constant of an 8×8 magic square

**261** is the number of essentially different ways to dissect a 16-gon into 7 quadrilaterals

**262** is the 5^{th} meandric number and the 9^{th} open meandric number

**263** is the largest known prime whose square is strobogrammatic

**264** is the largest known number whose square is undulating

**265** is the number of derangements of 6 items (A001116)

**266** is the Stirling number of the second kind S(8,6)

**267** is the number of planar partitions of 12

**268** is the smallest number whose product of digits is 6 times the sum of its digits

**269** is the number of 6-octs

**270** is a harmonic divisor number

**271** is the smallest prime p so that p-1 and p+1 are divisible by cubes

**272** is the 4^{th} tangent number

**273** = 333 in base 9

**274** is the Stirling number of the first kind s(6,2)

**275** is the number of partitions of 28 in which no part occurs only once

**276** = 1^{5} + 2^{5} + 3^{5}

**277** is a Perrin number

**278** is the closest integer to 6^{π }

**279** is the maximum number of 8^{th} powers needed to sum to any number

**280** is the number of ways 18 people around a round table can shake hands in a non-crossing way, up to rotation

**281** is the sum of the first 14 primes

**282** is the number of planar partitions of 9

**283** = 2^{5} + 8 + 3^{5}

**284** is an amicable number

**285** is the number of binary rooted trees with 13 vertices

**286** is the number of rooted trees with 9 vertices (A000081)

**287** is the sum of consecutive primes in 3 different ways

**288** is the smallest non-palindrome non-square that when multiplied by its reverse is a square

**289** is a Friedman number

**290** has a base 3 representation that ends with its base 6 representation

**291** is the largest number that is not the sum of distinct non-trivial powers

**292** is the number of ways to make change for a dollar

**293** is the number of ways to stack 20 boxes in a line so that each box lies on the table or on a box next to 2 boxes

**294** is the number of planar 2-connected graphs with 7 vertices

**295** is a structured deltoidal hexacontahedral number

**296** is the number of partitions of 30 into distinct parts

**297** is a Kaprekar number (A006886)

**298** is a value of n so that n(n+3) is a palindrome

**299** is the maximum number of regions a cube can be cut into with 12 cuts

**300** is the largest possible score in bowling (A060853)

**301** is a 6-hyperperfect number

**302** is the number of ways to play the first 3 moves in checkers (A133046)

**303** is the number of bipartite graphs with 8 vertices

**304** is a primitive semiperfect number

**305** is a hexagonal prism number (A005915)

**306** is the number of 5-digit triangular numbers

**307** is a non-palindrome with a palindromic square

**308** is a heptagonal pyramidal number

**309** is the smallest number whose 5^{th} power contains every digit at least once

**310** = 1234 in base 6

**311** is a permutable prime

**312** = 2222 in base 5

**313** is the number of intersections when all the diagonals of a regular dodecagon are drawn

**314** is the smallest number that can be written as the sum of of 3 positive distinct squares in 6 ways

**315** = (4+3) × (4+1) × (4+5)

**316** has a digit product which is the digit sum of (3^{1})^{6}

**317** is a value of n for which one less than the product of the first n primes is prime

**318** is the number of unlabeled partially ordered sets on 6 elements (A000112)

**319** is the smallest number with the property that the partition with the largest product does not have a maximum number of parts

**320** is the maximum determinant of a binary 10 X 10 matrix

**321** is a Delannoy number

**322** is the 12^{th} Lucas number (A000032)

**323** is the smallest composite number n that divides the (n+1)^{st} Fibonacci number

**324** is the largest possible product of positive integers with sum 16

**325** is a 3-hyperperfect number

**326** is the number of permutations of some subset of 5 elements

**327** is the largest number n so that n, 2n, and 3n together contain every digit from 1-9 exactly once

**328** concatenated with its successor is square

**329** is the number of forests with 10 vertices

**330** = _{11}C _{4}

**331** is both a centered pentagonal number and a centered hexagonal number

**332** is the number of 2-connected graphs with 7 vertices

**333** is the number of 7-hexes

**334** is the number of trees on 13 vertices with diameter 7

**335** is the number of degree 12 irreducible polynomials over GF(2)

**336** = _{8}P _{3}

**337** is the number of different resistances that can be created in a circuit of 8 equal resistors

**338** is the smallest number for which both the number of divisors and the sum of its prime factors is a perfect number

**339** is the number of ways to divide 5 black and 5 white beads into piles

**340** is a value of n for which n! + 1 is prime

**341** is the smallest pseudoprime in base 2

**342** is the number of inequivalent binary linear codes of length 8

**343** is a strong Friedman number

**344** is the smallest number that can be written as the sum of a cube and a 7^{th} power in more than one way

**345** is half again as large as the sum of its proper divisors

**346** is a Franel number

**347** is a Friedman number

**348** is the smallest number whose 5^{th} power contains exactly the same digits as another 5^{th} power

**349** is a Tetranacci -like number starting from 1, 1, 1, and 1

**350** is the Stirling number of the second kind S(7,4)

**351** is the smallest number so that it and the surrounding numbers are all products of 4 or more primes

**352** is the number of different arrangements of 9 non-attacking queens on an 9×9 chessboard

**353** is the smallest number whose 4^{th} power can be written as the sum of four 4^{th} powers

**354** is the sum of the first four 4^{th} powers

**355** is the number of labeled topologies with 4 elements

**356** is the smallest happy number of height 6

**357** has a base 3 representation that ends with its base 7 representation

**358** has a base 3 representation that ends with its base 7 representation

**359** has a base 3 representation that ends with its base 7 representation

**360** is the number of degrees in a circle.

**360** is also the order of a non-cyclic simple group (A001034)

**361** is the number of intersections on a Go board (A000290)

**362** and its double and triple all use the same number of digits in Roman numerals .

**363** is a perfect totient number

**364** = _{14}C _{3}

**365** is the smallest number that can be written as a sum of consecutive squares in more than 1 way

**366** is the number of days in a leap year

**367** is the largest number whose square has strictly increasing digits

**368** is the number of ways to tile a 4×15 rectangle with the pentominoes

**369** is the number of octominoes (A000105)

**370** is a narcissistic number

**371** is a narcissistic number

**372** is a hexagonal pyramidal number

**373** is a permutable prime

**374** is the smallest number that can be written as the sum of 3 squares in 8 ways

**375** is a truncated tetrahedral number

**376** is an automorphic number

**377** is the 14^{th} Fibonacci number (A000045)

**378** is the maximum number of regions a cube can be cut into with 13 cuts

**379** is a value of n for which one more than the product of the first n primes is prime

**380** is the number of necklaces possible with 13 beads, each being one of 2 colors

**381** is a Kaprekar constant in base 2

**382** is the smallest number n with σ (n) = σ (n+3)

**383** is the number of Hamiltonian graphs with 7 vertices

**385** is the number of partitions of 18

**386** is the number of regions the complex plane is cut into by drawing lines between all pairs of 11^{th} roots of unity

**387** is the smallest number with sort-then-add persistence of 10

**388** is the maximum value of n so that there exist 6 denominations of stamps so that every postage from 1 to n can be paid for with at most 6 stamps

**389** is the smallest prime so that it and the next 3 primes are all equal to 1 (mod 4)

**390** is the number of partitions of 32 into distinct parts

**391** ???

**392** is a Kaprekar constant in base 5

**393** is the 7^{th} central trinomial coefficient

**394** is a Schröder number

**395** does not occur in its factorial in base 2

**396** is the number of 3×3 sliding puzzle positions that require exactly 11 moves to solve starting with the hole in a corner

**397** is a Cuban prime

**398** is the number of integers with complexity 22

**399** is a Lucas-Carmichael number (A006972)

**400** = 1111 in base 7 (A023000)

**401** is the number of connected planar Eulerian graphs with 9 vertices

**402** is the number of graphs with 8 vertices and 9 edges

**403** is the product of two primes which are reverses of each other

**404** is the number of sided 10-hexes with holes

**405** is a pentagonal pyramidal number

**406** is the number of ways to tile a 3×17 rectangle with 3×1 rectangles

**407** is a narcissistic number

**408** is the 8^{th} Pell number

**409** is the number of graphs with 8 vertices with clique number 2

**410** is the smallest number that can be written as the sum of 2 distinct prime powers in 2 ways

**411** is a member of the Fibonacci -type sequence starting with 1 and 4

**412** is the number of subsets of {1,2,3,...,11} that have a sum divisible by 5

**413** is a structured hexagonal diamond number

**414** is a value of n for which n^{4}, n^{5}, n^{6}, and n^{7} have the same digit sum

**415** is the 10^{th} Iccanobif number, where each term is the reverse of the sum of the previous two numbers

**416** is the number of subsets of the 15^{th} roots of unity that add to a real number

**417** is the smallest number so that it and the next 3 numbers have different numbers of distinct prime factors

**418** has the property that the sum of its prime factors is equal to the product of its digits

**419** is the number of ways to divide a 6×6 grid of points into two sets using a straight line

**420** is the smallest number divisible by 1 through 7 (A169825)

**421** is the number of commutative monoids of order 6

**422** is the smallest number whose 8^{th} power has 21 digits

**423** is a number that does not have any digits in common with its cube

**424** ???

**425** is the number of subsets of {1,2,3,...,11} that have an integer average

**426** is a stella octangula number

**427** is a value of n for which n! + 1 is prime

**428** has the property that its square is the concatenation of two consecutive numbers

**429** is the 7^{th} Catalan number

**430** is the number of necklaces possible with 6 beads, each being one of 4 colors

**431** is the index of a prime Fibonacci number

**432** = 4 × 3^{3} × 2^{2}

**433** is the index of a prime Fibonacci number

**434** is the smallest composite value of n for which σ (n) + 2 = σ (n+2)

**435** is the number of ordered partitions of 16 into distinct parts

**436** is the smallest number whose cube contains four 8's

**437** has a cube with the last 3 digits the same as the 3 digits before that

**438** = 666 in base 8

**439** is the smallest prime where inserting the same digit between every pair of digits never yields another prime

**440** is the number of permutations of 12 items that fix 9 elements

**441** is the smallest square which is the sum of 6 consecutive cubes

**442** is the number of planar partitions of 13

**443** is a value of n for which σ (n) is a repdigit

**444** is the largest known n for which there is a unique integer solution to a_{1}+ ... +a_{n} = (a_{1})...(a_{n})

**445** has a base 10 representation which is the reverse of its base 9 representation

**446** is the smallest number that can be written as the sum of 3 distinct squares in 8 ways

**447** is the smallest number of convex quadrilaterals formed by 15 points in general position

**448** is the number of 10-iamonds

**449** has a base 3 representation that begins with its base 7 representation

**450** is the number of 13-iamonds with holes

**451** is the smallest number whose reciprocal has period 10

**452** is the closest integer to 7^{π }

**453** is the only number n so that n, 2n, and 6n together contain every digit exactly once

**454** is the largest number known that cannot be written as a sum of 7 or fewer cubes .

**455** = _{15}C _{3}

**456** is the number of tournaments with 7 vertices

**457** is the index of a prime Euclid number

**458** is a number that does not have any digits in common with its cube

**459** is the smallest number n for which reverse(n) - n contains the same digits as n

**460** ???

**461** is the number of ways to stack 18 pennies in contiguous rows so that each penny lies on the table or on two pennies

**462** = _{11}C _{5}

**463** is the smallest prime so that it and the next 6 primes are all equal to 3 (mod 4)

**464** is the maximum number of regions space can be divided into by 12 spheres

**465** is a Kaprekar constant in base 2

**466** = 1234 in base 7

**467** has strictly increasing digits in bases 7, 9, and 10

**468** = 3333 in base 5

**469** is a value of n for which n! - 1 is prime

**470** has a base 3 representation that ends with its base 6 representation

**471** is the smallest number with the property that its first 4 multiples contain the digit 4

**472** is the number of ways to tile a 5×5 square with integer -sided squares

**473** is the largest known number whose square and 4^{th} power use different digits

**474** is a member of the Fibonacci -type sequence starting with 1 and 8

**475** has a square that is composed of overlapping squares of smaller numbers

**476** is the number of different products of subsets of the set {1, 2, 3, ... 11}

**477** is the smallest number whose cube contains four 3's

**478** is the 7^{th} Pell-Lucas number

**479** is the number of sets of distinct positive integers with mean 6

**480** is the smallest number which can be written as the difference of 2 squares in 8 ways

**481** is the number of conjugacy classes in the automorphism group of the 10 dimensional hypercube .

**482** is a number whose square and cube use different digits

**483** is the last 3-digit string in the decimal expansion of π

**484** is a palindrome in base 3 and in base 10

**485** is the number of categories with 6 morphisms and 2 objects

**486** is a Perrin number

**487** is the number of Hadamard matrices of order 28

**488** ???

**489** is an octahedral number

**490** is the number of partitions of 19

**491** is the smallest number n so that the largest prime factors of the numbers n through n+4 decrease

**492** is a Hexanacci number

**493** is a Lucas 7-step number

**494** is the number of unlabeled distributive lattices with 14 elements

**495** is the Kaprekar constant for 3-digit numbers

**496** is the 3^{rd} perfect number (A000396)

**497** is the number of graphs with 8 edges

**498** is the number of necklaces possible with 8 beads, each being one of 3 colors

**499** is the number of ways to place 26 points on a 13×13 grid so that no 3 points are on a line

**500** is the number of planar partitions of 10

**501** is the number of partitions of 5 items into ordered lists

**502** uses the same digits as φ (502)

**503** is the smallest prime which is the sum of the cubes of the first few primes

**504** = _{9}P _{3}

**505** = _{10}C _{5} + _{10}C _{0} + _{10}C _{5}

**506** is the sum of the first 11 squares

**507** is the number of rooted ternary trees with 10 vertices

**508** ???

**509** is the index of a prime Fibonacci number

**510** is the number of binary rooted trees with 14 vertices

**511** = 111111111 in base 2 (A000225)

**512** is the cube of the sum of its digits

**513** is the number of conjugacy classes of the alternating group A_{22}

**514** ???

**515** is the number of graphs on 6 vertices with no isolated vertices

**516** is the number of partitions of 32 in which no part occurs only once

**517** does not occur in its factorial in base 2

**518** = 5^{1} + 1^{2} + 8^{3}

**519** is the number of trees on 15 vertices with diameter 5

**520** is the number of ways to place 2 non-attacking kings on a 6×6 chessboard

**521** is the 13^{th} Lucas number (A000032)

**522** is the number of ways to place a non-attacking white and black pawn on a 6×6 chessboard

**523** is the smallest prime that is followed by 17 composite numbers

**524** is the number of 6-kings

**525** is a hexagonal pyramidal number

**526** is the number of ways to cut a 8×8 chessboard into 2 pieces with equal areas with a cut that only travels up and right

**527** is the smallest number n for which there do not exist 4 smaller numbers so that a_{1}! a_{2}! a_{3}! a_{4}! n! is square

**528** concatenated with its successor is square

**529** is the smallest number n so that the continued fraction for n/k contains no 2's for any 1 ≤ k ≤ n

**530** is the sum of the first 3 perfect numbers (A092336)

**531** is the smallest number with the property that its first 4 multiples contain the digit 1

**532** is a hendecagonal pyramidal number

**533** is the number of degree sequences for graphs with 5 vertices

**534** ???

**535** is a palindrome whose φ (n) is also palindromic

**536** is the number of solutions of the stomachion puzzle

**537** divides the sum of the cubes of the first 537 primes

**538** is the 10^{th} open meandric number

**539** is the number of multigraphs with 5 vertices and 9 edges

**540** is divisible by its reverse (A169824)

**541** is the number of orderings of 5 objects with ties allowed (A000670)

**542** is a member of the Fibonacci -type sequence starting with 3 and 8

**543** is a number whose square and cube use different digits

**544** is the generalized Catalan number C(14,3)

**545** has a base 3 representation that begins with its base 4 representation

**546** undulates in bases 3, 4, and 5

**547** is the smallest number that can not be written using 11 copies of 11 and the operations +, –, ×, and ÷

**548** is the maximum number of 9^{th} powers needed to sum to any number

**549** ???

**550** is a pentagonal pyramidal number

**551** is the number of trees with 12 vertices

**552** is the number of prime knots with 11 crossings

**553** is a Huay rhombic dodecahedral number

**554** is the number of self-dual planar graphs with 20 edges

**555** is a repdigit

**556** are the first 3 digits of 4^{556}

**557** ???

**558** divides the sum of the largest prime factors of the first 558 positive integers

**559** is a centered cube number

**560** = _{16}C _{3}

**561** is the smallest Carmichael number

**562** is the maximum number of regions a circle can be cut into by joining 11 points on the circumference with straight lines

**563** is the largest known Wilson prime

**564** is the number of 13-ominoes with a horizontal or vertical line of symmetry

**565** is a structured truncated octahedral number

**566** is the number of ways to place 24 points on a 12×12 grid so that no 3 points are on a line

**567** has the property that it and its square together use the digits 1-9 once

**568** is the smallest number whose 7^{th} power can be written as the sum of seven 7^{th} powers

**569** is the smallest number n for which the concatenation of n, (n+1), ... (n+30) is prime

**570** is the product of all the prime palindromic Roman numerals

**571** is the index of a prime Fibonacci number

**572** is the smallest number which has equal numbers of every digit in bases 2 and 3

**573** has the property that its square is the concatenation of two consecutive numbers

**574** is the maximum number of pieces a torus can be cut into with 14 cuts

**575** is a palindrome that is one less than a square

**576** is the number of 4×4 Latin squares

**577** is a Proth prime

**578** is the number of graphs with 7 vertices with clique number 3

**579** is the number of graphs with 7 vertices that have chromatic number 3

**580** is the 6^{th} central quadrinomial coefficient

**581** has a base 3 representation that begins with its base 4 representation

**582** is the number of antisymmetric relations on a 5 element set

**583** is the smallest number whose reciprocal has period 26

**584** is the number of ways to color the vertices of a triangle with 12 colors, up to rotation

**585** is a palindrome in base 2, base 8, and in base 10

**586** is the smallest number that appears in its factorial 6 times

**587** is the smallest number whose digit sum is larger than that of its cube

**588** is the number of possible rook moves on a 7×7 chessboard

**589** is a centered tetrahedral number

**590** is a value of n for which φ (n) + φ (n+1) divides σ (n) + σ (n+1)

**591** is the number of ways to stack 23 boxes in a line so that each box lies on the table or on a box next to 2 boxes

**592** evenly divides the sum of its rotations

**593** is a Leyland number

**594** = 1^{5} + 2^{9} + 3^{4}

**595** is the number of ways to tile a 3×18 rectangle with 3×1 rectangles

**596** is the number of Hamiltonian cycles of a 4×9 rectangle graph

**597** is a value of n for which n!!! + 1 is prime

**598** = 5^{1} + 9^{2} + 8^{3}

**599** is the smallest number whose digits add to 23

**600** and its reverse are both the averages of twin primes

**601** is the location of the first occurrence of 3 consecutive zeroes in the decimal digits of π

**602** are the first 3 digits of 5^{602}

**603** is the smallest number n so that n, n+1, and n+2 are all the product of a prime and the square of a prime

**604** and the two numbers before it and after it are all products of exactly 3 primes

**605** has a sum of digits equal to its largest prime factor

**606** is the first non-trivial number that is both 11-gonal and centered 11-gonal

**607** is the exponent of a Mersenne prime (A000043, A000668)

**608** is a number that does not have any digits in common with its cube

**609** is a strobogrammatic number

**610** is the smallest Fibonacci number that begins with 6

**611** ???

**612** is a number whose square and cube use different digits

**613** is the index of a prime Lucas number

**614** is the smallest number that can be written as the sum of 3 squares in 9 ways

**615** is the trinomial coefficient T(10,6)

**616** is a Padovan number

**617** = 1!^{2} + 2!^{2} + 3!^{2} + 4!^{2}

**618** is the number of ternary square-free words of length 15

**619** is a strobogrammatic prime

**620** is the number of sided 7-hexes

**621** is the number of ways to 9-color the faces of a tetrahedron

**622** ???

**623** is the number of inequivalent asymmetric Ferrers graphs with 23 points

**624** is the smallest number with the property that its first 5 multiples contain the digit 2

**625** is an automorphic number

**626** is a palindrome in base 5 and in base 10

**627** is the number of partitions of 20

**628** is the sum of the squares of 4 consecutive primes

**629** evenly divides the sum of its rotations

**630** is a triangular number , 3 times a triangular number , and 6 times a triangular number

**631** has a base 2 representation that begins with its base 5 representation

**632** is the number of triangles formed by connecting the diagonals of a regular octagon

**633** is the smallest number n whose 5^{th} root has a decimal part that begins with the digits of n

**634** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**635** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**636** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**637** = 777 in base 9

**638** is the number of fixed 5-kings

**639** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**640** = 16!!!!!!

**641** is the smallest prime factor of 2^{25}+1

**642** is the smallest number with the property that its first 6 multiples contain the digit 2

**643** is the largest prime factor of 123456

**644** is a Perrin number

**645** is the largest n for which 1+2+3+ ... +n = 1^{2}+2^{2}+3^{2}+ ... +k^{2} for some k

**646** is the number of connected planar graphs with 7 vertices

**647** ???

**648** is the smallest number whose decimal part of its 6^{th} root begins with the digits 1-9 in some order

**649** ???

**650** is the sum of the first 12 squares

**651** has a 4^{th} power that is the sum of four 4^{th} powers

**652** is the only known non-perfect number whose number of divisors and sum of smaller divisors are perfect

**653** is the only known prime for which 5 is neither a primitive root or a quadratic residue of 4n^{2}+1

**654** has a square that is the sum of a cube and 5^{th} power

**655** ???

**656** is a palindrome in base 3 and in base 10

**657** is the number of ways to tile a 4×22 rectangle with 4×1 rectangles

**658** is the number of triangles of any size contained in the triangle of side 13 on a triangular grid

**659** is an Eisenstein-Mersenne prime (A066408)

**660** is the order of a non-cyclic simple group

**661** is the largest prime factor of 8! + 1

**662** is the index of the smallest triangular number that contains the digits 1, 2, 3, 4, and 5

**663** is the generalized Catalan number C(15,3)

**664** is a value of n so that n(n+7) is a palindrome

**665** is a member of the Fibonacci -type sequence starting with 1 and 4

**666** is the largest rep-digit triangular number

**667** is the number of asymmetric trees with 16 vertices

**668** is the number of legal pawn moves in Chess

**669** is the number of unsymmetrical ways to dissect a regular 12-gon into 10 triangles

**670** is an octahedral number

**671** is a rhombic dodecahedral number

**672** is a multi-perfect number

**673** is a Tetranacci -like number starting from 1, 1, 1, and 1

**674** ???

**675** is the smallest order for which there are 17 groups

**676** is the smallest palindromic square number whose square root is not palindromic

**677** is the closest integer to 11^{e }

**678** is a member of the Fibonacci -type sequence starting with 1 and 7

**679** is the smallest number with multiplicative persistence 5

**680** is the smallest tetrahedral number that is also the sum of 2 tetrahedral numbers

**681** divides the sum of the first 681 composite numbers

**682** = _{11}C _{6} + _{11}C _{8} + _{11}C _{2}

**683** is a Wagstaff prime

**684** is the sum of 3 consecutive cubes

**685** ???

**686** is the number of partitions of 35 in which no part occurs only once

**687** is the closest integer to 8^{π }

**688** is a Friedman number

**689** is the smallest number that can be written as the sum of 3 distinct squares in 9 ways

**690** is the smallest number that can not be written as the sum of a triangular number , a cube , and a Fibonacci number

**691** is the smallest prime p for which x^{5} = x^{4} + x^{3} + x^{2} + x + 1 (mod p) has 5 solutions

**692** is a number that does not have any digits in common with its cube

**693** are the first 3 decimal digits of ln (2)

**694** is the number of different arrangements (up to rotation and reflection) of 7 non-attacking rooks on a 7×7 chessboard

**695** is the maximum number of pieces a torus can be cut into with 15 cuts

**696** is a palindrome n so that n(n+8) is also palindromic

**697** is a 12-hyperperfect number

**698** = 3^{2} + 4^{3} + 5^{4}

**699** is a value of n for which |cos(n)| is smaller than any previous integer

**700** is the number of symmetric 8-cubes

**701** = 1^{0} + 2^{1} + 3^{2} + 4^{3} + 5^{4}

**702** ???

**703** is a Kaprekar number (A006886)

**704** is the number of sided octominoes

**705** is the smallest Lucas pseudoprime

**706** ???

**707** is the smallest number whose reciprocal has period 12

**708** is the maximum value of n so that there exist 4 denominations of stamps so that every postage from 1 to n can be paid for with at most 12 stamps

**709** is the number of connected planar graphs with 9 edges

**710** is the number of connected graphs with 9 edges

**711** is the name of a chain of convenience stores

**712** is the largest number known that does not have any digits in common with its 8^{th} power

**713** is the number of commutative monoids of order 7 with 4 idempotents

**714** is the smallest number which has equal numbers of every digit in bases 2 and 5

**715** = _{13}C _{4}

**716** is the smallest number whose cube contains four 6's

**717** is a palindrome in base 2 and in base 10

**718** is the number of unlabeled topologies with 6 elements

**719** is the number of rooted trees with 10 vertices (A000081)

**720** = 6!

**721** is the smallest number which can be written as the difference of 2 cubes in 2 ways

**722** is the sum of the 4^{th} powers of the first 3 primes

**723** = (1!)! + (2!)! + (3!)!

**724** is the number of different arrangements of 10 non-attacking queens on an 10×10 chessboard

**725** ???

**726** is a pentagonal pyramidal number

**727** has the property that its square is the concatenation of two consecutive numbers

**728** is the smallest number n where n and n+1 are both products of 5 or more primes

**729** = 3^{6}

**730** is the number of connected bipartite graphs with 9 vertices

**731** is the number of planar partitions of 14

**732** = 1^{7} + 2^{6} + 3^{5} + 4^{4} + 5^{3} + 6^{2} + 7^{1}

**733** is the sum of the digits of 4^{44}

**734** is the smallest number that can be written as the sum of 3 distinct non-zero squares in 10 ways

**735** is the smallest number that is the concatenation of its distinct prime factors

**736** is a strong Friedman number

**737** is a Boeing plane

**738** = 6 + 66 + 666

**739** has a base 2 representation that begins with its base 9 representation

**740** is the number of self-avoiding walks of length 8

**741** is the number of multigraphs with 6 vertices and 8 edges

**742** is the smallest number that is one more than triple its reverse

**743** is the number of independent sets of the graph of the 4-dimensional hypercube

**744** is the number of perfect squared rectangles of order 14

**745** is the smallest number whose square begins with three 5's

**746** = 1^{7} + 2^{4} + 3^{6}

**747** is a Boeing plane

**748** is the number of 3×3 sliding puzzle positions that require exactly 12 moves to solve starting with the hole in a corner

**749** is the number of ways to divide a 7×7 grid of points into two sets using a straight line

**750** is the Stirling number of the second kind S(10,8)

**751** is the index of a prime Woodall number

**752** is the number of conjugacy classes in the automorphism group of the 11 dimensional hypercube .

**753** is the smallest number whose cube contains 4 consecutive 7's

**754** ???

**755** is the number of trees on 14 vertices with diameter 6

**756** is the maximum number of regions space can be divided into by 14 spheres

**757** is the smallest number whose reciprocal has a period of 27

**758** ???

**759** is the number of octads in the large Witt design

**760** is the number of partitions of 37 into distinct parts

**761** ???

**762** is the starting location of 999999 in the decimal expansion of π

**763** is the smallest number whose 4^{th} power contains every digit at least once

**764** is the number of 8×8 symmetric permutation matrices

**765** is a Kaprekar constant in base 2

**766** is the number of series-reduced planted trees with 9 leaves

**767** is the largest n so that n^{2} = _{m}C _{0} + _{m}C _{1} + _{m}C _{2} + _{m}C _{3} has a solution

**768** is the number of subsets of {1,2,3,...,12} that have an integer average

**769** is the total number of digits of all binary numbers of length 1-7

**770** is the number of digits of the 15^{th} perfect number (A061193)

**771** is the number of intersections when all the diagonals of a regular 14-gon are drawn

**772** ???

**773** is the smallest odd number n so that n+2^{k} is composite for all k<n

**774** ???

**775** is the smallest number whose 9^{th} power has 26 digits

**776** ???

**777** is a repdigit in base 6 and in base 10

**778** is the number of ways a 5×1 rectangle can be surrounded by 5×1 rectangles

**779** ???

**780** = (5+7) × (5+8) × (5+0)

**781** = 11111 in base 5

**782** is a number whose sum of divisors is a 4^{th} power

**783** is the number of 11-ominoes that tile the plane by translation

**784** is the sum of the first 7 cubes

**785** are the last 3 digits of the sum of the first 785 squares

**786** is the largest known n for which _{2n}C _{n} is not divisible by the square of an odd prime

**787** is a palindrome in base 3 and in base 10

**788** is the smallest of 6 consecutive numbers divisible by 6 consecutive primes

**789** are the first 3 digits of 9^{789}

**790** ???

**791** is the smallest number n where either it or its neighbors are divisible by the numbers from 1 to 12

**792** is the number of partitions of 21

**793** is one less than twice its reverse

**794** = 1^{6} + 2^{6} + 3^{6}

**795** is a number whose sum of divisors is a 4^{th} power

**796** ???

**797** is the number of functional graphs on 9 vertices

**798** is the number of ternary square-free words of length 16

**799** is the smallest number whose sum of digits is composite and whose sum of digits cubed is prime

**800** = 2222 in base 7 (A125725)

**801** = (7! + 8! + 9! + 10!) / (7 × 8 × 9 × 10)

**802** is the number of isomers of C_{13}H_{28}

**803** is a value of n for which σ (n) is a repdigit

**804** is a value of n for which 2φ (n) = φ (n+1)

**805** is the number of possible positions in Checkers after 4 moves

**806** is not the sum of a square , a cube , a 4^{th} power, and a 5^{th} power

**807** ???

**808** is a strobogrammatic number

**809** is a member of the Fibonacci -type sequence starting with 1 and 5

**810** is a value of n for which n-1 and n+1 are twin primes , and so are 2n-1 and 2n+1

**811** ???

**812** is the number of triangles of any size contained in the triangle of side 14 on a triangular grid

**813** are the first 3 digits of 813^{e }

**814** is a value of n so that n(n+5) is a palindrome

**815** is a Lucas 3-step number

**816** = _{18}C _{3}

**817** ???

**818** is the number of ways to dissect a 12-gon using non-crossing diagonals into polygons with an even number of sides.

**819** is the smallest number so that it and its successor are both the product of 2 primes and the square of a prime

**820** = 1111 in base 9

**821** is a number n for which n, n+2, n+6, and n+8 are all prime

**822** is the number of planar graphs with 7 vertices

**823** is a number that does not have any digits in common with its cube

**824** ???

**825** is the number of ways to legally add 2 sets of parentheses to a product of 9 variables

**826** ???

**827** is the number of asymmetric trees with 11 vertices

**828** ???

**829** is a value of n for which π(n) is the product of the digits of n

**830** ???

**831** is the number of monic polynomials of degree 9 with integer coefficients whose complex roots are all in the unit disk

**832** is the maximum number of pieces a torus can be cut into with 16 cuts

**833** is a centered octahedral number

**834** is the maximum number of regions a cube can be cut into with 17 cuts

**835** is the 9^{th} Motzkin number

**836** is a non-palindrome with a palindromic square

**837** ???

**838** ???

**839** has a base 5 representation that begins with its base 9 representation

**840** is the smallest number divisble by 1 through 8

**841** is a square that is also the sum of 2 consecutive squares

**842** is the ratio of Fibonacci numbers

**843** is the 14^{th} Lucas number (A000032)

**844** is the smallest number so that it and the next four numbers are squareful numbers

**845** ???

**846** has the property that its square is the concatenation of two consecutive numbers

**847** is the sum of the digits of the 14^{th} Mersenne prime (A066538)

**848** is the number of inequivalent binary linear codes of length 9

**849** is a value of n for which σ (n-1) = σ (n+1)

**850** is the number of trees on 14 vertices with diameter 7

**851** is the number of ordered partitions of 18 into distinct parts

**852** is the number of 6-colorable connected graphs with 7 vertices

**853** is the number of connected graphs with 7 vertices (A001349)

**854** has the property that it and its square together use the digits 1-9 once

**855** is the smallest number which is the sum of 5 consecutive squares or 2 consecutive cubes

**856** is a member of the Fibonacci -type sequence starting with 1 and 9

**857** is a value of n for which φ (n) = φ (n-1) + φ (n-2)

**858** is the smallest palindrome with 4 different prime factors

**859** is the number of planar partitions of 11 (A000219)

**860** ???

**861** = 7 + 77 + 777

**862** is a number whose sum of divisors is a 4^{th} power

**863** is a value of n so that n(n+6) is a palindrome

**864** is the number of partitions of 38 into distinct parts (A000009)

**865** ???

**866** is the number of sided 10-iamonds (A006534)

**867** is the number of graphs with 8 vertices that have chromatic number 5

**868** has a square root whose decimal part starts with the digits 1-9 in some order

**869** is the number of different resistances that can be created in a circuit of 9 equal resistors (A048211)

**870** is the sum of its digits and the cube of its digits

**871** ???

**872** is a value of n for which n! + 1 is prime (A002981)

**873** = 1! + 2! + 3! + 4! + 5! + 6! (A007489)

**874** is the number of positive integer solutions to (1 + 1/a)(1 + 1/b)(1 + 1/c)(1 + 1/d)(1 + 1/e) = 2

**875** is 3-automorphic

**876** is a dodecagonal pyramidal number (A007587)

**877** is the 7^{th} Bell number (A000110)

**878** is the number of 3×3 sliding puzzle positions that require exactly 29 moves to solve starting with the hole on a side

**879** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**880** is the number of 4×4 magic squares (A006052)

**881** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**882** is the smallest number whose square begins with three 7's

**883** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**884** is a number n whose 5^{th} root has a decimal part that begins with the digits of n

**885** is an enneagonal pyramidal number (A007584)

**886** ???

**887** is a value of n for which σ (n) is a repdigit

**888** and the following 18 numbers are composite

**889** is a Kaprekar constant in base 2

**890** ???

**891** is the number of unlabeled distributive lattices with 15 elements

**892** is the smallest integer ratio of a 13-digit number to its product of digits

**893** has a square whose digits each occur twice

**894** has a base 5 representation that begins with its base 9 representation

**895** is a Woodall number (A003261)

**896** is not the sum of 4 non-zero squares

**897** is a Cullen number (A002064)

**898** is a member of the Fibonacci -type sequence starting with 2 and 5

**899** is the product of twin primes

**900** has a base 5 representation that begins with its base 9 representation

**901** is the sum of the digits of the first 100 positive integers

**902** is a value of n so that n(n+7) is a palindrome

**903** is the 6^{th} super Catalan number

**904** has a cube that is the sum of 3 positive cubes

**905** is the smallest composite number that is not the sum of a prime and a power of 2

**906** is the number of perfect graphs with 7 vertices

**907** is the largest n so that **Q** (√n) has class number 3

**908** ???

**909** is a value of n that has has no digits in common with 2n, 3n, 4n, 5n, 6n, 7n, 8n, or 9n

**910** is the generalized Catalan number C(11,4)

**911** is the American emergency number

**912** is a Pentanacci number

**913** has exactly the same digits in 3 different bases

**914** is the number of binary rooted trees with 15 vertices

**915** ???

**916** is a strobogrammatic number

**917** is the only positive number known whose 9^{th} power can be written as the sum of ten 9^{th} powers

**918** is a number that does not have any digits in common with its cube

**919** is the smallest number which is not the difference between palindromes

**920** is a truncated cube number

**921** ???

**922** = 1234 in base 9

**923** multiplied by its successor gives a number concatenated with itself

**924** is the 6^{th} central binomial coefficient

**925** is the number of partitions of 37 in which no part occurs only once

**926** is the smallest number that can not be formed using the digits 1-6 at most once, with the operators +, –, ×, ÷, and ^

**927** is the 13^{th} tribonacci number

**928** ???

**929** is a Proth prime

**930** is the number of even permutations on 7 elements with no fixed points

**931** ???

**932** ???

**933** is a house number

**934** has a 5^{th} root that starts 3.25252225...

**935** is a Lucas-Carmichael number (A006972)

**936** is a pentagonal pyramidal number

**937** ???

**938** ???

**939** has a cube root whose decimal part starts with the digits 1-9 in some order

**940** is the maximum number of regions space can be divided into by 15 spheres

**941** is the smallest number which is the reverse of the sum of its proper substrings

**942** is the smallest number whose cube contains five 8's

**943** is a Lucas 6-step number

**944** ???

**945** is the smallest odd abundant number (A005101, A005231)

**946** is a hexagonal pyramidal number

**947** ???

**948** is the number of symmetric plane partitions of 24

**949** is the larger number in a Ruth-Aaron pair

**950** is the generalized Catalan number C(17,3)

**951** is the number of functions from 8 unlabeled points to themselves

**952** = 9^{3} + 5^{3} + 2^{3} + 9 × 5 × 2

**953** is the largest prime factor of 54321

**954** ???

**955** is the number of ways to to arrange the numbers 1-9 around a circle so that the sums of adjacent numbers are distinct

**956** is the number of multigraphs with 16 vertices and 4 edges

**957** is a value of n for which σ (n) = σ (n+1)

**958** is the number of labeled 3-colorable graphs with 5 vertices

**959** is a Carol number (A093112)

**960** is the sum of its digits and the cube of its digits (A065138)

**961** is a square whose digits can be rotated to give another square (A035126)

**962** ???

**963** is a value of n for which π(n) is the product of the digits of n (A097220)

**964** is the number of 3×3 sliding puzzle positions that require exactly 12 moves to solve starting with the hole in the center (A089474)

**965** ???

**966** is the Stirling number of the second kind S(8,3) (A008277, A048993, A000392)

**967** is the number of 6-digit triangular numbers (A068094)

**968** is an Achilles number (A052486)

**969** is a tetrahedral palindrome (A006030)

**970** ???

**971** ???

**972** is an Achilles number (A052486)

**973** is the number of inequivalent asymmetric Ferrers graphs with 25 points (A000701)

**974** is the number of multigraphs with 5 vertices and 10 edges (A014395)

**975** is the number of 11-ominoes that contain 1 hole (A057418)

**976** has a square formed by inserting a block of digits inside itself (A052212)

**977** is a Stern prime (A042978)

**978** 2^{4} + 3^{4} + 4^{4} + 5^{4}

**979** is the sum of the first five 4^{th} powers (A000538)

**980** ???

**981** is the smallest number that has 5 different partitions into 3 parts with the same product

**982** is the number of partitions of 39 into distinct parts (A000009)

**983** is a Wedderburn-Etherington number (A001190)

**984** = 8 + 88 + 888 (A099675)

**985** is the 9^{th} Pell number (A000129)

**986** is a strobogrammatic number (A000787)

**987** is the 16^{th} Fibonacci number (A000045)

**988** is the maximum number of regions a cube can be cut into with 18 cuts (A000125)

**989** is the smallest number such that it and its reverse are divisible by 43 (A055483)

**990** is a triangular number that is the product of 3 consecutive integers (A001219)

**991** is a permutable prime (A003459, A129338)

**992** is the number of differential structures on the 11-dimensional hypersphere (A001676)

**993** is the number of paraffins with 8 carbon atoms (A005961)

**994** is the smallest number with the property that its first 18 multiples contain the digit 9 (A039940)

**995** has a square formed by inserting a block of digits inside itself (A046829)

**996** has a square formed by inserting a block of digits inside itself (A046829)

**997** has a cube root that starts 9.98998998...

**998** is the smallest number with the property that its first 55 multiples contain the digit 9 (A039940)

**999** is a Kaprekar number (A006886)