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# Annotated version of "What's Special About This Number?" (Part 6)

## 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.

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 6: The Numbers 6000 to 6999

**6000** is the number of subsets of the 24^{th} roots of unity that add to 1

**6001** has a cube that is a concatenation of other cubes

**6002** is the number of digits of the 24^{th} Mersenne prime (A028335)

**6003** has a square with the first 3 digits the same as the next 3 digits

**6006** is the number of intersections when all the diagonals of a regular 21-gon are drawn

**6008** = _{14}C _{6} + _{14}C _{0} + _{14}C _{0} + _{14}C _{8}

**6009** is a strobogrammatic number

**6011** is a member of the Fibonacci -type sequence starting with 3 and 8

**6012** has a square with the last 3 digits the same as the 3 digits before that

**6014** has a square that is formed by 3 squares that overlap by 1 digit

**6016** is the maximum number of pieces a torus can be cut into with 32 cuts

**6017** is a centered octahedral number

**6018** is the maximum number of regions a cube can be cut into with 33 cuts

**6020** is the number of Hamiltonian graphs with 8 vertices

**6021** has a square that is formed by 3 squares that overlap by 1 digit

**6024** is a value of n for which |cos(n)| is smaller than any previous integer

**6025** are the last 4 digits of the sum of the first 6025 squares

**6032** is the number of ways to place 2 non-attacking knights on a 9×9 chessboard

**6035** is a number whose sum of divisors is a 5^{th} power

**6040** is the number of ways to divide 6 couples into pairs where no pair is a couple

**6048** is the order of a non-cyclic simple group

**6050** has a sum of digits equal to its largest prime factor

**6058** is a number that does not have any digits in common with its cube

**6065** is the closest integer to 16^{π }

**6070** is a structured truncated tetrahedral number

**6072** is the order of a non-cyclic simple group

**6073** is the order of a non-cyclic simple group

**6075** is an Achilles number

**6077** has a square with the last 3 digits the same as the 3 digits before that

**6080** is the smallest number n>1 whose base 14 representation is equal to φ (n)

**6081** has a cube that is the sum of 3 positive cubes

**6083** has a square that is the sum of a cube and a 4^{th} power

**6084** is the sum of the first 12 cubes

**6092** is the number of 16-ominoes with a line of symmetry

**6093** is a value of n for which 3n and 5n together use each digit exactly once

**6095** is a rhombic dodecahedral number

**6097** is an hexagonal prism number

**6099** concatenated with its successor is square

**6100** has the property that if each digit is replaced by its square , the resulting number is a square

**6102** is the largest number n known where φ (n) is the reverse of n

**6105** is a Huay rhombic dodecahedral number

**6106** is a value of n for which 2φ (n) = φ (n+1)

**6107** is a Perrin number

**6111** is a value of n for which σ (n-1) = σ (n+1)

**6119** is a strobogrammatic number

**6120** is a highly abundant number (A002093)

**6121** is the smallest number whose cube contains 4 consecutive 3's

**6128** is a betrothed number

**6137** is a centered dodecahedral number

**6138** is the number of quasi-tetrominoes that fit inside a 7×7 grid

**6141** is a Kaprekar constant in base 2

**6142** is the number of inequivalent asymmetric Ferrers graphs with 34 points

**6143** is the smallest prime that contains twelve 1's in binary

**6144** = 16!!!!

**6145** is a Friedman number

**6155** is a member of the Fibonacci -type sequence starting with 2 and 5

**6164** is the number of 11-ominoes that tile the plane using 180 degree rotations

**6167** has a 4^{th} power that is the sum of four 4^{th} powers

**6168** is the number of inequivalent Ferrers graphs with 34 points

**6170** = 5 + 55 + 555 + 5555

**6171** has the property that dropping its first and last digits gives its largest prime factor

**6173** is a prime that remains prime if any digit is deleted

**6174** is the Kaprekar constant for 4-digit numbers

**6175** is the number of regions formed when all diagonals are drawn in a regular 21-gon

**6176** is the last 4-digit sequence to appear in the decimal expansion of π

**6179** is a value of n for which 4n and 5n together use each digit exactly once

**6180** is the smallest number n with φ (n) = 2 reverse(n)

**6181** is an octahedral number

**6187** is a Smith brother

**6188** = _{17}C _{5}

**6189** is the number of ways to write 17 as an ordered sum of positive integers , where adjacent numbers are different

**6194** is the number of ways to place a non-attacking white and black pawn on a 10×10 chessboard

**6196** is the number of regions the complex plane is cut into by drawing lines between all pairs of 21^{st} roots of unity

**6197** is a narcissistic number in base 6

**6200** is a harmonic divisor number

**6201** is the sum of the first 26 squares

**6210** is the number of 5×5 matrices with non-negative entries with every row and column adding to 2

**6211** is a Cuban prime

**6216** has a square with the first 3 digits the same as the next 3 digits

**6219** is a value of n for which 4n and 5n together use each digit exactly once

**6220** = 44444 in base 6

**6221** = 666 + 2222 + 2222 + 1111

**6222** is the smallest number that can not be written as the sum of 2 triangular numbers and a power of 2

**6223** = 666 + 2222 + 2 + 3333

**6224** is the number of permutations of 8 elements have 4^{th} power equal to the identity element

**6225** = 666 + 2 + 2 + 5555

**6232** is an amicable number

**6237** is a number whose sum of the squares of its divisors is a square

**6239** , followed by 6239 7's, is prime

**6240** is a highly abundant number (A002093)

**6244** is a member of the Fibonacci -type sequence starting with 2 and 9

**6245** is the smallest number whose square contains 4 consecutive internal 0's

**6248** is the smallest number with the property that its first 8 multiples contain the digit 4

**6249** is the smallest number with the property that its first 10 multiples contain the digit 4

**6250** is a Leyland number

**6256** is a hendecagonal pyramidal number

**6257** is the number of essentially different ways to dissect a 20-gon into 9 quadrilaterals

**6266** is a truncated octahedral number

**6267** is the number of 15-iamonds with holes

**6270** is a value of n for which n-1 and n+1 are twin primes , and so are 2n-1 and 2n+1

**6271** is the smallest number requiring an addition chain of length 17

**6272** is the number of ways to tile a 4×29 rectangle with 4×1 rectangles

**6273** is the number of ways to 9-color the vertices of a pentagon, up to rotations and reflections

**6274** has a cube whose digits occur with the same frequency

**6276** is a value of n for which φ (n) = φ (reverse(n))

**6279** is the number of subsequences of {1,2,3,...14} in which every odd number has an even neighbor

**6280** has a sum of prime factors that is equal to the sum of the prime factors of the two preceding numbers

**6290** is the number of 13-iamonds that do not tile the plane

**6293** is the number of ordered partitions of 24 into distinct parts

**6296** has a square with the first 3 digits the same as the next 3 digits

**6297** is a value of n for which n and 5n together use each digit 1-9 exactly once

**6299** is the smallest number with complexity 30

**6300** is divisible by its reverse

**6307** is the largest n so that **Q** (√n) has class number 8

**6309** is the closest integer to 25^{e }

**6310** is the smallest number whose 5^{th} power has 19 digits

**6312** is the sum of its proper divisors that contain the digit 5

**6318** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**6320** is the Entringer number E(8,4).

**6322** is the number of idempotent functions from a set of 7 elements into itself.

**6327** = 324 + 325 + . . . + 342 = 343 + 344 + . . . + 360

**6331** has the same digits as the 6331^{st} prime

**6336** is the number of ways to tile a 9×4 rectangle with 2×1 rectangles

**6343** is the number of quasi-triominoes that fit inside a 14×14 grid

**6347** has the same digits as the 6347^{th} prime

**6348** is a pentagonal pyramidal number

**6351** is the largest number known that is not the sum of 3 squares or cubes

**6354** is the number of 14-iamonds that tile the plane

**6360** is a value of n for which n-1 and n+1 are twin primes , and so are 3n-1 and 3n+1

**6368** is an amicable number

**6371** has a square that is the sum of 2 relatively prime cubes

**6374** is a value of n for which 4n and 5n together use each digit exactly once

**6375** has a square with the first 3 digits the same as the next 3 digits

**6378** is the number of partitions of 55 into distinct parts

**6379** is a value of n for which |cos(n)| is smaller than any previous integer

**6380** is a value of n for which n! + 1 is prime

**6381** is the smallest value of n for which n and 9n together use each digit 1-9 exactly once

**6384** is an icosahedral number

**6385** is the number of ways to stack 18 pennies in a line so that each penny lies on the table or on two pennies

**6389** is the number of functional graphs on 11 vertices

**6391** is a hexagonal pyramidal number

**6395** is the number of ways to divide a 12×12 grid of points into two sets using a straight line

**6396** is a divisor of the sum of the 4^{th} powers of its divisors

**6397** has the same digits as the 6397^{th} prime

**6399** and its successor are both divisible by 4^{th} powers

**6400** is a square whose digits are non-increasing

**6403** has a square with the first 3 digits the same as the last 3 digits

**6404** is a value of n for which n!! - 1 is prime

**6406** is the number of permutations of 8 elements where every cycle has equal length

**6408** is the sum of the squares . of the first 13 primes

**6409** is a house number

**6411** is a truncated cube number

**6424** is the number of minimal covers of a set containing 6 elements

**6427** is the number of ways a 6×6 square can be tiled with 1×1 and 2×2 squares

**6432** has the same digits as the 6432^{nd} prime

**6434** is the number of divisors of the 18^{th} perfect number

**6435** = _{15}C _{7}

**6443** has a cube whose digits occur with the same frequency

**6444** is the smallest number whose 5^{th} power starts with 5 identical digits

**6445** , followed by 6445 1's, is prime

**6454** is the smallest value of n for which π(10n) = n

**6455** is the smallest value of n for which the n^{th} prime begins with the digits of n

**6456** is a value of n for which the n^{th} prime begins with the digits of n

**6457** is a value of n for which the n^{th} prime begins with the digits of n

**6458** would be prime if preceded and followed by a 1, 3, 7, or 9

**6459** is a value of n for which the n^{th} prime begins with the digits of n

**6460** is a value of n for which the n^{th} prime begins with the digits of n

**6462** divides the sum of the digits of 6462!

**6466** is the largest known value of n for which the n^{th} prime begins with the digits of n

**6471** is a value of n for which n and 9n together use each digit 1-9 exactly once

**6472** is the number of polyominoes with 9 or fewer squares

**6475** is a value of n for which π(n) is the product of the digits of n

**6479** is a value of n for which n and 8n together use each digit 1-9 exactly once

**6481** = (3^{12} + 1) / (3^{4} + 1)

**6487** is the number of partitions of 51 in which no part occurs only once

**6488** would be prime if preceded and followed by a 1, 3, 7, or 9

**6489** is half again as large as the sum of its proper divisors

**6490** is the number of ways to place 2 non-attacking bishops on a 11×11 chessboard

**6498** is the index of a triangular number containing only 3 different digits

**6500** is a number n whose sum of the factorials of its digits is equal to π(n)

**6501** has a square whose reverse is also a square

**6505** is the number of 9-hexes without holes

**6506** is a value of n for which the first n binary digits of π form a prime

**6510** is a number n whose sum of the factorials of its digits is equal to π(n)

**6511** is a number n whose sum of the factorials of its digits is equal to π(n)

**6514** is the sum of the 4^{th} powers of the digits of the sum of the 4^{th} powers of the digits of itself

**6517** has a sum of digits equal to its largest prime factor

**6521** is a number n whose sum of the factorials of its digits is equal to π(n)

**6523** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**6524** has the property that its square starts with its reverse

**6525** is a centered icosahedral number

**6526** is the smallest number whose 10^{th} power contains exactly the same digits as another 10^{th} power

**6527** is a value of n for which φ (n) = φ (n-1) + φ (n-2)

**6529** is a Proth prime

**6532** is a member of the Fibonacci -type sequence starting with 1 and 6

**6533** is the number of digits of the 25^{th} Mersenne prime (A028335)

**6534** is a value of n for which 3n and 7n together use each digit exactly once

**6537** is the smallest value of n for which the numbers n-6 through n+6 can not be written as the sum of 2 squares

**6540** is the number of terms in the 17^{th} derivative of f(f(f(x)))

**6543** has a square root that has four 8's immediately after the decimal point

**6544** is a number n whose 9^{th} root has a decimal part that begins with the digits of n

**6545** and its reverse are tetrahedral numbers

**6547** is the number of binary 4×4 matrices with no row or column containing 3 consecutive 1's

**6552** is the number of different full houses in 5 card poker with one joker

**6553** is a Lucas 5-step number

**6556** is the largest palindrome that can be made using 5 digits and the 4 arithmetic operations

**6557** is the product of two consecutive primes

**6560** is the smallest number n where n and n+1 are both products of 7 or more primes

**6561** = 3^{8}

**6569** is a value of n for which one less than the product of the first n primes is prime

**6572** is the number of 9-hexes

**6576** = (6! - 6) + (5! - 5) + (7! - 7) + (6! - 6)

**6578** is the smallest number which can be written as the sum of three 4^{th} powers in 2 ways

**6579** is the number of ways to color the vertices of a triangle with 27 colors, up to rotation

**6580** is the maximum number of regions a cube can be cut into with 34 cuts

**6581** has the same digits as the 6581^{st} prime

**6583** is a value of n for which σ (φ (n)) = 2σ (n)

**6588** is the number of sided 12-iamonds

**6593** = 6 + 5555 + 999 + 33

**6594** is a value of n for which 5n and 7n together use each digit exactly once

**6596** has a square comprised of the digits 0-7

**6601** is a Carmichael number

**6603** is a number whose square and cube use different digits

**6608** is the maximum number of regions space can be divided into by 28 spheres

**6609** has a 4^{th} power that is the sum of four 4^{th} powers

**6611** is a value of n such that the n^{th} Cullen number is prime

**6615** is an odd abundant number (A005101, A005231)

**6620** is the number of 11-ominoes that tile the plane

**6623** has the property that the sum of its prime factors is equal to the product of its digits

**6630** is the number of triangles of any size contained in the triangle of side 29 on a triangular grid

**6636** has exactly the same digits in 3 different bases

**6639** divides 1^{1} + 2^{2} + 3^{3} + ^{ . . .} + 6639^{6639}

**6642** can be written as the sum of 2 or 4 positive 4^{th} powers

**6643** is the smallest number which is palindromic in bases 2 and 3

**6647** has a sum of digits equal to its largest prime factor

**6651** is the index of a triangular number containing only 3 different digits

**6653** , when concatenated with 4 less than itself, is square

**6654** is the smallest number whose decimal part of its 4^{th} root starts with the digits 0-9 in some order

**6663** is a value of n for which σ (n) is a repdigit

**6665** is a centered tetrahedral number

**6666** is a repdigit

**6667** is the number of self-dual planar graphs with 24 edges

**6668** is the number of trees on 21 vertices with diameter 5

**6669** is the sum of 3 consecutive cubes

**6680** = 6666 + 6 + 8 + 0

**6681** = 6666 + 6 + 8 + 1

**6682** = 6666 + 6 + 8 + 2

**6683** = 6666 + 6 + 8 + 3

**6684** = 6666 + 6 + 8 + 4

**6685** = 6666 + 6 + 8 + 5

**6686** = 6666 + 6 + 8 + 6

**6687** = 6666 + 6 + 8 + 7

**6688** = 6666 + 6 + 8 + 8

**6689** = 6666 + 6 + 8 + 9

**6694** is a value of n for which the sum of the first n primes is square

**6699** is a strobogrammatic number

**6700** has a cube that contains the digits 6700 in reverse order

**6704** is the number of rooted 8-hexes

**6706** is the number of Hamiltonian paths in a 8×5 rectangle graph

**6712** is the index of a triangular number containing only 3 different digits

**6714** is the index of a triangular number containing only 3 different digits

**6716** is the 4-digit string that appears latest in the decimal expansion of π

**6720** = _{8}P _{5}

**6721** is a composite value of n that divides the (n-1)^{st} Fibonacci number

**6723** is a value of n for which 3n and 8n together use each digit exactly once

**6726** is the 10^{th} Pell-Lucas number

**6728** is the number of domino tilings of a 6×6 square

**6729** is the smallest value of n for which n and 2n together use each digit 1-9 exactly once

**6731** would be prime if preceded and followed by a 1, 3, 7, or 9

**6732** is a value of n for which _{2n}C _{n} is divisible by n^{2}

**6734** is a value of n for which |cos(n)| is smaller than any previous integer

**6735** is a stella octangula number

**6736** is the number of 3×3 sliding puzzle positions that require exactly 17 moves to solve starting with the hole in the center

**6740** is the number of 13-iamonds that do not tile the plane

**6741** is a value of n for which n and 8n together use each digit 1-9 exactly once

**6742** has a square where the first 6 digits alternate

**6743** is the number of binary 4×5 matrices with no consecutive 1's in any row or column

**6745** 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 25 stamps

**6751** is the number of digits of the 23^{rd} perfect number (A061193)

**6754** is the smallest number in base 9 to have 5 different digits

**6756** has a cube that is the sum of 3 positive cubes

**6759** is the number of graphs with 10 vertices and 11 edges

**6764** is the sum of the first 18 Fibonacci numbers

**6765** is the 20^{th} Fibonacci number

**6768** has a 9^{th} root that starts 2.664444666...

**6769** is the Stirling number of the first kind s(8,4)

**6772** has a square whose digits each occur twice

**6779** = 6666 + 7 + 7 + 99

**6780** has the same digits as the 6780^{th} prime

**6786** is a triangular number whose internal digits are triangular and whose external digits are triangular

**6788** is the smallest number with multiplicative persistence 6

**6789** is the largest 4-digit number with increasing digits

**6791** is a value of n for which n and 8n together use each digit 1-9 exactly once

**6792** is a value of n for which n and 2n together use each digit 1-9 exactly once

**6793** is the smallest prime so that it and the next 2 primes all end in 3

**6794** has the property that dropping its first and last digits gives its largest prime factor

**6797** is a number whose sum of divisors is a 5^{th} power

**6801** has a 4^{th} power that is the sum of four 4^{th} powers

**6802** is the number of ways to move a rook from corner to opposite corner on a 6×6 chessboard

**6811** is not the sum of a square , a cube , a 4^{th} power, and a 5^{th} power

**6813** is the smallest number whose 6^{th} power has 24 digits

**6816** is the index of a triangular number containing only 3 different digits

**6818** = 1^{8} + 2^{8} + 3^{8}

**6819** = 20457 / 3, and each digit is contained in the equation exactly once

**6820** is the number of regions formed when all diagonals are drawn in a regular 23-gon

**6822** uses the same digits as φ (6822)

**6825** is an odd primitive abundant number (A091191, A006038)

**6828** is the number of ways to start with a knight in the corner of an 8×8 chessboard, make 8 moves, and end on the same square

**6831** is a structured truncated octahedral number

**6837** is the number of 8-digit squares

**6839** is a value of n for which n and 8n together use each digit 1-9 exactly once

**6840** is the number of ways to place 2 non-attacking kings on a 11×11 chessboard

**6842** is the number of partitions of 31

**6845** would be prime if preceded and followed by a 1, 3, 7, or 9

**6849** is a value of n for which 2n and 3n together use each digit exactly once

**6850** is the smallest value of n for which n, n+1, n+2, n+3, n+4, and n+5 have the same number of prime factors

**6853** is a value of n for which n, n+1, n+2, and n+3 have the same number of divisors

**6859** = 19^{3}

**6860** is a heptagonal pyramidal number

**6861** is a value of n for which σ (n-1) + σ (n+1) = σ (2n)

**6863** is a prime that is the sum of the square of a prime and the cube of a prime

**6864** = 6666 + 88 + 66 + 44

**6865** is the maximum value of n so that there exist 5 denominations of stamps so that every postage from 1 to n can be paid for with at most 17 stamps

**6867** can be written as the sum of 2, 3, 4, or 5 positive cubes

**6868** is the larger number in a Ruth-Aaron pair

**6874** is equal to the sum of its anti-divisors

**6875** is 3-automorphic

**6879** is the number of planar partitions of 15

**6880** is a vampire number

**6886** is a palindrome in base 9 and in base 10

**6888** has a square with 3/4 of the digits are the same

**6889** is a strobogrammatic square

**6895** is a value of n for which 2n and 7n together use each digit exactly once

**6896** has a square root whose decimal part starts with the digits 0-9 in some order

**6900** is the number of ways to place 2 non-attacking knights on a 11×11 chessboard

**6902** is the number of Hamiltonian paths of a 3×10 rectangle graph

**6903** is a value of n for which σ (n-1) = σ (n+1)

**6905** has a 5^{th} root whose decimal part starts with the digits 1-9 in some order

**6912** = 6 × 9 × 1 × 2^{7}

**6917** is a value of n for which n! - 1 is prime

**6918** = 20754 / 3, and each digit is contained in the equation exactly once

**6919** is the number of non-invertible knots with 13 crossings

**6922** is the number of polycubes containing 8 cubes

**6924** is the magic constant of a 24×24 magic square

**6926** has a square whose digits each occur twice

**6927** is a value of n for which n and 2n together use each digit 1-9 exactly once

**6928** is the number of inequivalent binary linear codes of length 11

**6930** is the square root of a triangular number

**6931** has the same digits as the 6931^{st} prime

**6935** is the smallest number whose cube contains six 3's

**6936** is the number of ways to legally add 2 sets of parentheses to a product of 16 variables

**6939** is a value of n for which 3n and 5n together use each digit exactly once

**6940** is the sum of its proper divisors that contain the digit 3

**6941** has a square whose digits each occur twice

**6942** is the number of labeled topologies with 5 elements

**6944** is the number of degree sequences for graphs with 6 vertices

**6949** is the smallest number that can not be written as the sum of 3 volumes of rectangular boxes with integer dimensions less than 16

**6951** has exactly the same digits in 3 different bases

**6952** = 1738 × 4 and each digit from 1-9 is contained in the equation exactly once

**6953** = 66 + 999 + 5555 + 333

**6954** is the trinomial coefficient T(19,15)

**6956** is the number of triangles formed by drawing all diagonals of a regular 12-gon

**6960** is the number of ways to place 2 non-attacking queens on a 10×10 chessboard

**6966** is the number of planar graphs with 8 vertices

**6969** is a strobogrammatic number

**6972** is the number of possible positions in Checkers containing 2 checkers

**6976** is the number of binary 5×5 matrices A with the property that A^{2}=0 (mod 2)

**6982** is a value of n for which the sum of the first n composite number numbers is a square

**6983** is the smallest prime that can only be made into 1 other prime by changing a single digit

**6984** can be written as the sum of 2, 3, 4, or 5 positive cubes

**6985** is the smallest number that can be written as the sum of 3 or more consecutive squares , or as the sum of 3 or more consecutive cubes

**6987** is the number of digits of the 26^{th} Mersenne prime (A028335)

**6989** has the property that the concatenation of its prime factors in increasing order is a square

**6991** is a value of n for which reverse(φ (n)) = φ (reverse(n))

**6996** is a palindrome n so that n(n+8) is also palindromic

**6998** is a member of the Fibonacci -type sequence starting with 4 and 9

**6999** is the smallest number whose digits add to 33