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

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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:

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 24th 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 24th 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 = 14C 6 + 14C 0 + 14C 0 + 14C 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 5th 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 16π


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 4th 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 4th power that is the sum of four 4th 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 = 17C 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 21st 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 4th 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 25e


6310 is the smallest number whose 5th 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 6331st 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 6347th 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 4th powers of its divisors


6397 has the same digits as the 6397th prime


6399 and its successor are both divisible by 4th 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 6432nd prime


6434 is the number of divisors of the 18th perfect number

6435 = 15C 7

6443 has a cube whose digits occur with the same frequency

6444 is the smallest number whose 5th 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 nth prime begins with the digits of n

6456 is a value of n for which the nth prime begins with the digits of n

6457 is a value of n for which the nth 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 nth prime begins with the digits of n

6460 is a value of n for which the nth 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 nth 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 = (312 + 1) / (34 + 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 4th powers of the digits of the sum of the 4th 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 10th power contains exactly the same digits as another 10th 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 25th 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 17th 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 9th 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 = 38

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 4th 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 6581st 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 4th power that is the sum of four 4th powers

6611 is a value of n such that the nth 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 11 + 22 + 33 + . . . + 66396639


6642 can be written as the sum of 2 or 4 positive 4th 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 4th 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 = 8P 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 10th 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 2nC n is divisible by n2

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 23rd 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 20th Fibonacci number

6768 has a 9th 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 6780th 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 5th power


6801 has a 4th power that is the sum of four 4th 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 4th power, and a 5th power

6813 is the smallest number whose 6th power has 24 digits


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


6818 = 18 + 28 + 38


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 = 193

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 5th root whose decimal part starts with the digits 1-9 in some order


6912 = 6 × 9 × 1 × 27

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 6931st 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 A2=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 26th 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

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