This site is supported by donations to The OEIS Foundation.

# Annotated version of "What's Special About This Number?" (Part 8)

## 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 8: The Numbers 8000 to 8999

**8000** is the smallest cube which is also the sum of 4 consecutive cubes

**8001** is a Kaprekar constant in base 2

**8002** is the index of a triangular number containing only 3 different digits

**8003** has the property that if each digit is replaced by its square , the resulting number is a square

**8004** has a square with the first 3 digits the same as the next 3 digits

**8008** = _{16}C _{6}

**8010** uses the same digits as π (8010)

**8012** is the number of 3-connected planar maps with 18 edges

**8016** has a square with the last 3 digits the same as the 3 digits before that

**8022** uses the same digits as φ (8022)

**8026** is the number of planar partitions of 19

**8042** is the largest number known which cannot be written as a sum of 7 or fewer cubes

**8043** has a square whose digits each occur twice

**8045** is the number of 6-digit twin primes
**8051** is the number of partitions of 52 in which no part occurs only once

**8056** is the number of triangles of any size contained in the triangle of side 31 on a triangular grid

**8064** = (1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9) / (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)

**8071** is the number of connected graphs with 11 edges

**8074** is the trinomial coefficient T(12,6)

**8077** is a value of n for which n^{2} and n^{3} use the same digits

**8080** has a square root that has four 8's immediately after the decimal point

**8082** has a square comprised of the digits 1-8

**8083** is a value of n for which n concatenated with n-2 is square

**8085** is an odd primitive abundant number (A091191, A006038)

**8087** is a Lucas 9-step number

**8089** is the pseudosquare modulo 13

**8090** is a Perrin number

**8092** is a Friedman number

**8100** is divisible by its reverse

**8103** is the closest integer to e ^{9}

**8104** is equal to the sum of its anti-divisors

**8118** is a strobogrammatic number

**8119** is an NSW number

**8121** is the smallest number whose cube contains seven 5's

**8125** is the smallest number that can be written as the sum of 2 squares in 5 ways

**8128** is the 4^{th} perfect number

**8129** is a member of the Fibonacci -type sequence starting with 2 and 7

**8135** is the 7^{th} central pentanomial coefficient

**8136** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**8149** is a value of n for which 2n and 7n together use each digit exactly once

**8152** is the number of symmetric arrangements of 8 non-attacking queens on a 8×8 chessboard

**8154** is a value of n for which |cos(n)| is smaller than any previous integer

**8156** has a cube that is only 24 away from a square

**8165** has a square that begins with four 6's

**8169** = 24507 / 3, and each digit is contained in the equation exactly once

**8170** is an enneagonal pyramidal number

**8174** is a value of n for which n and 8n together use each digit 1-9 exactly once

**8176** is a stella octangula number

**8178** is the number of ways 13 people can line up so that only one person has a taller person in front of him

**8179** is a value of n for which 4n and 5n together use each digit exactly once

**8180** is the maximum number of regions space can be divided into by 30 spheres

**8184** has exactly the same digits in 3 different bases

**8189** is the index of a triangular number containing only 3 different digits

**8190** is a harmonic divisor number

**8191** is a Mersenne prime (A000043, A000668)

**8192** is the smallest non-trivial 13^{th} power

**8194** is the number of subsets of the 26^{th} roots of unity that add to 0

**8195** is the number of 17-ominoes with a horizontal or vertical line of symmetry

**8196** has a square whose digits each occur twice

**8198** is the index of a Fibonacci number whose first 9 digits are the digits 1-9 rearranged

**8200** = 8 + 2^{13} + 0 + 0

**8201** = 8 + 2^{13} + 0 + 1

**8202** = 8 + 2^{13} + 0 + 2

**8203** = 8 + 2^{13} + 0 + 3

**8204** = 8 + 2^{13} + 0 + 4

**8205** = 8 + 2^{13} + 0 + 5

**8206** = 8 + 2^{13} + 0 + 6

**8207** = 8 + 2^{13} + 0 + 7

**8208** is a narcissistic number

**8209** = 8 + 2^{13} + 0 + 9

**8217** is a centered icosahedral number

**8219** is a value of n for which 4n and 5n together use each digit exactly once

**8220** and its reverse are both the averages of twin primes

**8221** has a base 3 representation that begins with its base 6 representation

**8225** are the first 4 digits of 8^{8225}

**8226** is the sum of its proper divisors that contain the digit 4

**8229** has a square whose digits each occur twice

**8230** is the number of necklaces with 8 beads, each one of 4 colors

**8241** is a value of n for which n has σ (n) / reverse(n) divisors

**8242** , when concatenated with one less than it, is square

**8256** is the number of different arrangements (up to rotation and reflection) of 30 non-attacking bishops on a 16×16 chessboard

**8257** is the sum of the squares . of the first 14 primes

**8258** is the number of different positions in Connect Four after 6 moves

**8265** has a 7^{th} root whose decimal part starts with the digits 1-9 in some order

**8269** is a Cuban prime

**8280** is the smaller number in a Ruth-Aaron pair

**8281** is the only 4-digit square whose two 2-digit pairs are consecutive

**8283** has a base 8 representation which is the reverse of its base 7 representation

**8292** is the number of anisohedral 22-iamonds

**8294** has the property that dropping its first and last digits gives its largest prime factor

**8299** is a value of n for which reverse(φ (n)) = φ (reverse(n))

**8303** = 12345 in base 9

**8304** is the number of subsets of the 18^{th} roots of unity that add to a real number

**8305** has the same digits as the 8305^{th} prime

**8313** is a dodecagonal pyramidal number

**8316** is the sum of 3 consecutive cubes

**8320** is the number of subsets of {1, 1/2, 1/3, ... 1/42} that sum to an integer

**8321** is a Poulet number

**8338** is a value of n so that n(n+4) is a palindrome

**8340** is a value of n so that (n-1)^{2} + n^{2} + (n+1)^{2} is a palindrome

**8342** is the number of partitions of 53 in which no part occurs only once

**8345** is the smallest number in base 6 to have 6 different digits

**8349** is the number of partitions of 32

**8350** is the trinomial coefficient T(10,1)

**8351** has the same digits as the 8351^{st} prime

**8353** is the smallest number whose 4^{th} power contains 5 consecutive 6's

**8355** has the same digits as the 8355^{th} prime

**8360** has a square whose digits each occur twice

**8361** is a Leyland number

**8363** is the number of 5-digit primes

**8368** has a 6^{th} power whose first few digits are 34334444...

**8369** is the largest prime factor of 2 × 3 × 5 × 7 × 11 × 13 × 17 - 1

**8372** is a hexagonal pyramidal number

**8373** has a 4^{th} power that is the sum of four 4^{th} powers

**8375** is the smallest number which has equal numbers of every digit in bases 2 and 6

**8378** has a 10^{th} root whose decimal part starts with the digits 1-9 in some order

**8379** is a value of n for which 5n and 8n together use each digit exactly once

**8382** is the index of a triangular number containing only 3 different digits

**8384** is the maximum number of 13^{th} powers needed to sum to any number

**8385** is a structured great rhombicubeoctahedral number

**8388** and its reverse are both the averages of twin primes

**8390** is the number of linear spaces on 7 labeled points

**8392** is a value of n for which n, n+1, n+2, and n+3 have the same number of divisors

**8393** is a value of n for which σ (reverse(n)) = φ (n)

**8394** is a value of n for which n and 8n together use each digit 1-9 exactly once

**8396** does not occur in its factorial in base 2

**8397** is the largest known composite number n so that _{3n}C _{n} = 3^{n} (mod n)

**8398** is the 10^{th} super-ballot number

**8400** is the number of legal queen moves in Chess

**8401** has the property that if each digit is replaced by its square , the resulting number is a square

**8403** = 33333 in base 7

**8406** is the number of ways to divide 8 black and 8 white beads into piles

**8408** has 8408 / π(8408) divisors

**8411** would be prime if preceded and followed by a 1, 3, 7, or 9

**8415** is an odd primitive abundant number (A091191, A006038)

**8418** is the number of necklaces possible with 11 beads, each being one of 3 colors

**8419** is a value of n for which n and 8n together use each digit 1-9 exactly once

**8420** is the number of symmetric ways to fold a strip of 20 stamps

**8421** = 1111 in base 20

**8428** is the number of quasi-triominoes that fit inside a 15×15 grid

**8430** and its reverse are both the averages of twin primes

**8433** has a 4^{th} power that is the sum of four 4^{th} powers

**8436** = _{38}C _{3}

**8439** is a value of n for which n and 8n together use each digit 1-9 exactly once

**8440** is a truncated square pyramid number

**8441** is the sum of the cubes of 3 consecutive primes

**8442** is the smallest value of n for which the numbers n-7 through n+7 can not be written as the sum of 2 squares

**8451** is the number of 3×3 matrices in base 3 with determinant 0

**8455** is the trinomial coefficient T(20,16)

**8459** is a value of n so that n(n+4) is a palindrome

**8461** is the smallest number whose 9^{th} power starts with 5 identical digits

**8463** is the smaller number in a Ruth-Aaron pair

**8464** is the number of different products of subsets of the set {1, 2, 3, ... 17}

**8465** = 4^{3} + 5^{4} + 6^{5}

**8467** has a 9^{th} root whose decimal part starts with the digits 1-9 in some order

**8469** is a value of n for which 2n and 3n together use each digit exactly once

**8470** is the number of conjugacy classes in the automorphism group of the 17 dimensional hypercube .

**8472** is the maximum number of pieces a torus can be cut into with 36 cuts

**8473** is a centered octahedral number

**8474** is the maximum number of regions a cube can be cut into with 37 cuts

**8475** is the first of four consecutive squareful numbers

**8477** = 1^{0} + 2^{1} + 3^{2} + 4^{3} + 5^{4} + 6^{5}

**8481** is a Poulet number

**8484** is the reciprocal of the sum of the reciprocals of 13332 and its reverse

**8486** = 888 + 44 + 888 + 6666

**8492** is the number of arrangements of 5 non-attacking queens on a 11×5 chessboard

**8493** has a 4^{th} power that is the sum of four 4^{th} powers

**8494** is a value of n for which σ (n) = φ (n) + φ (n-1) + φ (n-2)

**8497** is the number of anisohedral 17-hexes

**8499** is the sum of the squares of 3 consecutive primes

**8505** = 21!!!!!!

**8506** is the number of isomers of C_{13}H_{26} without any double bonds

**8509** is a value of n for which |cos(n)| is smaller than any previous integer

**8510** is a value of n for which the sum of the first n primes is a palindrome

**8512** is the number of non-intersecting rook paths joining opposite corners of a 5×5 chessboard

**8515** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**8517** has a 4^{th} power that is the sum of four 4^{th} powers

**8521** is a prime that is the average of two 4^{th} powers

**8523** is the first of four consecutive squareful numbers

**8525** has a square whose digits each occur twice

**8526** is a Rhonda number

**8533** has the property that dropping its first and last digits gives its largest prime factor

**8538** is the sum of its proper divisors that contain the digit 4

**8541** is a value of n so that n(n+6) is a palindrome

**8545** is the number of ways to stack 36 boxes in a line so that each box lies on the table or on a box next to 2 boxes

**8547** is a divisor of 111111.

**8548** is the sum of the squares of 4 consecutive primes

**8549** has the property that the sum of its proper divisors is the sum of the squares of its digits

**8555** is the sum of the first 29 squares

**8558** is a Schröder number

**8559** has a square comprised of the digits 1-8

**8562** is the sum of its proper divisors that contain the digit 4

**8563** is the index of a triangular number containing only 3 different digits

**8568** = _{18}C _{5}

**8569** is a centered dodecahedral number

**8571** shares 3 consecutive digits with one of its prime factors

**8575** is an Achilles number

**8576** can be written as the sum of 2, 3, 4, or 5 positive cubes

**8577** has a 4^{th} power that is the sum of four 4^{th} powers

**8578** appears inside its 4^{th} power

**8579** divides 1^{1} + 2^{2} + 3^{3} + ^{ . . .} + 8579^{8579}

**8580** is the number of subsets of the 28^{th} roots of unity that add to 1

**8582** is the number of monoids of order 7 with 5 idempotents

**8586** has exactly the same digits in 3 different bases

**8599** is the number of forests with 14 vertices

**8602** is the generalized Catalan number C(20,4)

**8610** = 400 + 401 + . . . + 420 = 421 + 422 + . . . + 440

**8614** and its prime factors contain every digit from 1-9 exactly once

**8626** is the number of asymmetric trees with 13 vertices

**8627** is a value of n for which 2n and 7n together use each digit exactly once

**8631** is a value of n for which 3n and 7n together use each digit exactly once

**8633** is the product of two consecutive primes

**8637** has a 4^{th} power that is the sum of four 4^{th} powers

**8638** = 7 + 77 + 777 + 7777

**8640** = 2! × 3! × 6!

**8641** is the number of ways to tile a 3×25 rectangle with 3×1 rectangles

**8642** has digits in arithmetic sequence

**8646** divides 2^{8646} + 2

**8649** is a value of n for which 2n and 7n together use each digit exactly once

**8657** is the number of ways to tile a 4×30 rectangle with 4×1 rectangles

**8658** is the sum of the first 4 perfect numbers

**8663** has the property that if each digit is replaced by its square , the resulting number is a square

**8664** = 888 + 6666 + 666 + 444

**8666** has a 9^{th} root whose decimal part starts with the digits 1-9 in some order

**8669** 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 18 stamps

**8670** is a value of n for which n!! - 1 is prime

**8672** is the number of 14-ominoes that tile the plane by translation

**8680** has a base 5 representation that ends with its base 7 representation

**8681** has a base 5 representation that ends with its base 7 representation

**8682** has a base 5 representation that ends with its base 7 representation

**8683** has a base 5 representation that ends with its base 7 representation

**8684** has a base 5 representation that ends with its base 7 representation

**8688** is the number of possible configurations of pegs (up to symmetry) after 26 jumps in solitaire

**8695** is a centered tetrahedral number

**8697** is a structured octagonal anti-diamond number

**8698** is a strobogrammatic number

**8703** has a cube that is the sum of 3 positive cubes

**8712** is 4 times its reverse

**8714** is the number of ways 24 people around a round table can shake hands in a non-crossing way, up to rotation

**8718** is the smallest n for which Σ_{k≤n} 1/(k ln k) ≥ 3

**8721** is a value of n for which φ (n) and σ (n) are square

**8732** has exactly the same digits in 3 different bases

**8736** is the smallest number that appears in its factorial 10 times

**8739** is a permutation of the sum of its proper divisors

**8743** is a number whose sum of divisors is a 4^{th} power

**8744** is the number of subsets of {1,2,3,...,17} that have a sum divisible by 15

**8745** is the number of ways to divide a 13×13 grid of points into two sets using a straight line

**8748** is the largest number whose prime factors add to 25

**8751** is a perfect totient number

**8753** = 88 + 7777 + 555 + 333

**8758** = 88 + 7777 + 5 + 888

**8761** is the number of ordered partitions of 25 into distinct parts

**8763** and its successor have the same digits in their prime factorization

**8765** has digits in arithmetic sequence

**8771** 2^{4} + 3^{4} + 4^{4} + 5^{4} + 6^{4} + 7^{4} + 8^{4}

**8772** is the sum of the first eight 4^{th} powers

**8778** is both a triangular number and 3 times a triangular number

**8779** is is the largest prime factor of 100000000001

**8781** is the closest integer to 18^{π }

**8784** is a value of n for which 2n and 5n together use each digit exactly once

**8785** is the number of 13-iamonds without holes

**8788** is an Achilles number

**8793** is a value of n for which n!!! - 1 is prime

**8796** is a value of n for which 5n and 7n together use each digit exactly once

**8797** is a structured hexagonal diamond number

**8801** is the magic constant of a 26×26 magic square

**8808** is the number of partitions of 58 into distinct parts

**8810** has a square whose digits each occur twice

**8813** is the number of chiral invertible knots with 14 crossings

**8814** is the number of multigraphs with 27 vertices and 4 edges

**8816** is a value of n for which reverse(φ (n)) = φ (reverse(n))

**8819** is the smallest number whose square begins with four 7's

**8820** is a highly abundant number [A002093)

**8821** has the property that if each of its digits is replaced by its cube , the result is a square

**8826** is the sum of its proper divisors that contain the digit 4

**8829** is a value of n for which 6n and 7n together use each digit exactly once

**8831** would be prime if preceded and followed by a 1, 3, 7, or 9

**8833** = 88^{2} + 33^{2}

**8835** is the index of a triangular number containing only 3 different digits

**8837** is the smallest number that can not be written as the sum of 3 volumes of rectangular boxes with integer dimensions less than 17

**8838** and its reverse are both the averages of twin primes

**8840** is the number of triangles of any size contained in the triangle of side 32 on a triangular grid

**8843** is the smallest number that can not be written as the sum of 2 volumes of rectangular boxes with integer dimensions less than 22

**8846** is the number of divisors of the 20^{th} perfect number

**8854** is the number of possible rows in a 20×20 crossword puzzle

**8855** is a Lucas-Carmichael number (A006972)

**8856** is the number of subsets of {1,2,3,...,16} that have an integer average

**8857** is a structured truncated tetrahedral number

**8860** is the smallest number n so that n+3, n^{2}+3^{2}, n^{4}+3^{4}, and n^{8}+3^{8} are all prime

**8864** is a value of n for which |cos(n)| is smaller than any previous integer

**8867** is the smallest prime with multiplicative persistence 6

**8874** has a square that is the concatenation of two consecutive even numbers

**8878** is the number of intersections when all the diagonals of a regular 23-gon are drawn

**8883** does not occur in its factorial in base 2

**8887** is a value of n for which σ (n) is a repdigit

**8888** is a repdigit

**8892** is a betrothed number

**8902** is the number of possibilities for the first 1.5 moves in Chess

**8905** multiplied by its successor gives a number concatenated with itself

**8910** is divisible by its reverse

**8911** is a Carmichael number

**8913** 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 27 stamps

**8922** is the sum of its proper divisors that contain the digit 4

**8923** is the numerator of 1 / 1^{1} + 1 / 2^{2} + 1 / 3^{3} + 1 / 4^{4}

**8925** is an odd primitive abundant number (A091191, A006038)

**8930** = 8888 + 9 + 33 + 0

**8931** = 8888 + 9 + 33 + 1

**8932** = 8888 + 9 + 33 + 2

**8933** = 8888 + 9 + 33 + 3

**8934** = 8888 + 9 + 33 + 4

**8935** = 8888 + 9 + 33 + 5

**8936** = 8888 + 9 + 33 + 6

**8937** = 8888 + 9 + 33 + 7

**8938** = 8888 + 9 + 33 + 8

**8939** = 8888 + 9 + 33 + 9

**8942** is a value of n for which n and 8n together use each digit 1-9 exactly once

**8944** is the sum of the cubes of the first 7 primes

**8950** has a 4^{th} root whose decimal part starts with the digits 1-9 in some order

**8953** is the 10^{th} central trinomial coefficient

**8954** is the first of four consecutive squareful numbers

**8958** has a 4^{th} power whose product of digits is also a 4^{th} power

**8959** is the smallest multiple of 31 whose digits add to 31

**8964** is the smallest number with the property that its first 6 multiples contain the digit 8

**8965** is a value of n for which n^{2} and n^{3} use the same digits

**8968** is a strobogrammatic number

**8970** = 8 + 9^{4} + 7^{4} + 0

**8971** = 8 + 9^{4} + 7^{4} + 1

**8972** = 8 + 9^{4} + 7^{4} + 2

**8973** = 8 + 9^{4} + 7^{4} + 3

**8974** = 8 + 9^{4} + 7^{4} + 4

**8975** = 8 + 9^{4} + 7^{4} + 5

**8976** = 8 + 9^{4} + 7^{4} + 6

**8977** = 8 + 9^{4} + 7^{4} + 7

**8978** = 8 + 9^{4} + 7^{4} + 8

**8979** = 8 + 9^{4} + 7^{4} + 9

**8980** is a value of n for which the first n binary digits of π form a prime

**8982** uses the same digits as φ (8982)

**8989** is a Delannoy number

**8991** is the smallest number so that it and its successor are both the product of a prime and the 5^{th} power of a prime

**8993** is a Huay rhombic dodecahedral number

**8999** is the smallest number whose digits add to 35