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 = ₁₆C 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 n² and n³ 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 + 2¹³ + 0 + 0

8201 = 8 + 2¹³ + 0 + 1

8202 = 8 + 2¹³ + 0 + 2

8203 = 8 + 2¹³ + 0 + 3

8204 = 8 + 2¹³ + 0 + 4

8205 = 8 + 2¹³ + 0 + 5

8206 = 8 + 2¹³ + 0 + 6

8207 = 8 + 2¹³ + 0 + 7

8208 is a narcissistic number

8209 = 8 + 2¹³ + 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 8⁸²²⁵

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 + n² + (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 _3nC _n = 3ⁿ (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 = ₃₈C 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 = 4³ + 5⁴ + 6⁵

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 = 1⁰ + 2¹ + 3² + 4³ + 5⁴ + 6⁵

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 C₁₃H₂₆ 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 = ₁₈C 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 1¹ + 2² + 3³ + ^{. . .} + 8579⁸⁵⁷⁹

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 2⁸⁶⁴⁶ + 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 2⁴ + 3⁴ + 4⁴ + 5⁴ + 6⁴ + 7⁴ + 8⁴

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 = 88² + 33²

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, n²+3², n⁴+3⁴, and n⁸+3⁸ 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 / 1¹ + 1 / 2² + 1 / 3³ + 1 / 4⁴

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 n² and n³ use the same digits

8968 is a strobogrammatic number

8970 = 8 + 9⁴ + 7⁴ + 0

8971 = 8 + 9⁴ + 7⁴ + 1

8972 = 8 + 9⁴ + 7⁴ + 2

8973 = 8 + 9⁴ + 7⁴ + 3

8974 = 8 + 9⁴ + 7⁴ + 4

8975 = 8 + 9⁴ + 7⁴ + 5

8976 = 8 + 9⁴ + 7⁴ + 6

8977 = 8 + 9⁴ + 7⁴ + 7

8978 = 8 + 9⁴ + 7⁴ + 8

8979 = 8 + 9⁴ + 7⁴ + 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