Annotated version of "What's Special About This Number?" (Part 5)

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 5: The Numbers 5000 to 5999

5000 is the largest number whose English name does not repeat any letters

5001 appears inside its 4^th power

5002 has a 4^th power containing only 4 different digits

5005 is the smallest palindromic product of 4 consecutive primes

5009 would be prime if preceded and followed by a 1, 3, 7, or 9

5010 has a square with the last 3 digits the same as the 3 digits before that

5016 is a heptagonal pyramidal number

5020 is an amicable number

5024 is a member of the Fibonacci -type sequence starting with 2 and 7

5026 is the number of connected graphs with 11 vertices and 1 cycle

5030 is the closest integer to 23^e

5036 and the two numbers before it and after it are all products of exactly 3 primes

5039 is the number of planar partitions of 18

5040 = 7!

5041 is the largest square known of the form n! + 1

5042 is the number of subsets of {1,2,3,...,16} that have a sum divisible by 13

5044 is a value of n for which φ (n) and σ (n) are square

5046 is the first of five consecutive squareful numbers

5048 is the number of strongly connected digraphs with 5 vertices

5049 is an octagonal pyramidal number (A002414)

5050 is the sum of the first 100 integers (A000217)

5054 = 555 + 0 + 55 + 4444

5055 has exactly the same digits in 3 different bases

5056 is the number of ways to flip a coin 13 times and get at least 3 heads in a row

5057 is the number of squares in a 16×16 grid of squares with diagonals drawn

5059 is the number of inequivalent asymmetric Ferrers graphs with 33 points

5061 is a number n whose 5^th root has a decimal part that begins with the digits of n

5069 is the number of square-free graphs with 10 vertices

5071 is a Lucas 3-step number and a Lucas 4-step number

5077 has a square whose digits each occur twice

5078 is the number of rectangles with corners on an 12×12 grid of points

5080 is a structured truncated octahedral number

5083 is an centered icosahedral number

5084 is the number of inequivalent Ferrers graphs with 33 points

5087 has an eleventh root whose decimal part starts with the digits 1-9 in some order

5088 divides the sum of the digits of 2⁵⁰⁸⁸ × 5088!

5096 is the number of possible rook moves on a 14×14 chessboard

5098 is the number of 3-valent trees with 17 vertices

5100 is divisible by its reverse

5103 and its successor are both divisible by 4^th powers

5104 is the smallest number that can be written as the sum of 3 cubes in 3 ways

5105 would be prime if preceded and followed by a 1, 3, 7, or 9

5107 preceded by 5107 1's is prime

5108 is the number of different flushes in 5 card poker

5109 is the number of conjugacy classes of the alternating group A₃₃

5118 is the maximum value of n so that there exist 6 denominations of stamps so that every postage from 1 to n can be paid for with at most 12 stamps

5120 is the number of edges in a 10 dimensional hypercube

5130 is a value of n for which φ (n) and σ (n) are square

5133 is the smallest integer ratio of a 18-digit number to its product of digits

5134 has the property that the sum of the factorials of its digits is its largest prime factor

5135 is not the sum of a square , a cube , a 4^th power, and a 5^th power

5136 does not occur in its factorial in base 2

5141 is the sum of the first 17 numbers that have digit sum 17

5141 is the only four digit number that is reversed in hexadecimal

5142 is the sum of its proper divisors that contain the digit 7

5143 = 555 + 111 + 4444 + 33

5146 has a base 3 representation that begins with its base 7 representation

5152 is the number of legal rook moves in Chess

5153 is an Eisenstein-Mersenne prime (A066408)

5160 is a hendecagonal pyramidal number

5161 = 5! + (1+6)! + 1!

5162 = 5! + (1+6)! + 2

5163 = 5! + (1+6)! + 3

5164 = 5! + (1+6)! + 4

5165 = 5! + (1+6)! + 5

5166 = 5! + (1+6)! + 6

5167 = 5! + (1+6)! + 7

5168 has a square root that has four 8's immediately after the decimal point

5169 = 5! + (1+6)! + 9

5172 has a cube whose last few digits are ...48848448

5174 has a 4^th power containing only 4 different digits

5176 is the number of labeled graphs with 6 vertices that have chromatic number 2

5177 is the number of labeled bipartite graphs with 6 vertices

5180 is the smallest number whose 7^th power has 26 digits

5181 is a structured octagonal anti-diamond number

5182 is a number whose sum of divisors is a 5^th power

5183 is the product of twin primes

5184 is the number of ways to place 2 non-attacking rooks on a 9×9 chessboard

5185 is the number of 2×2 singular matrices mod 17

5186 is equal to the sum of its anti-divisors

5187 is the only number n known for which φ (n-1) = φ (n) = φ (n+1)

5191 is a value of n for which σ (n+1) = 2σ (n)

5199 divides the sum of the cubes of the first 5199 primes

5200 is divisible by its reverse

5204 has the property that if each digit is replaced by its square , the resulting number is a square

5211 has a square root whose decimal part starts with the digits 1-9 in some order

5216 is a structured hexagonal diamond number

5218 is the number of 3-colorable graphs connected graphs with 8 vertices

5220 = 1111 in base 17

5222 has the property that the sum of the n^th powers of its digits is prime for 1 ≤ n &\le 9

5225 is the number of ways to color the vertices of a triangle with 25 colors, up to rotation

5226 is the number of ways to color the vertices of a square with 12 colors, up to rotation

5229 uses the same digits as φ (5229)

5234 has a cube that is only 17 away from a square

5237 is a value of n for which n and 8n together use each digit 1-9 exactly once

5239 has a square whose digits each occur twice

5241 is the starting location of 7777 in the decimal expansion of π

5242 is the number of ways to place 8 non-attacking kings on a 8×8 chessboard so that there is a king in every row and column

5244 is the sum of consecutive squares in 2 ways

5247 is the number of ordered ways to write 1 as a sum of reciprocals of integers no larger than 10

5248 is the number of ordered ways to write 1 as a sum of reciprocals of integers no larger than 11

5250 is the number of linear geometries on 10 unlabeled points

5252 is the maximum number of regions space can be divided into by 26 spheres

5256 is the number of labeled partially ordered sets of 4 elements

5257 is a member of the Fibonacci -type sequence starting with 1 and 8

5258 has a base 8 representation which is the reverse of its base 7 representation

5260 is the number of multigraphs with 24 vertices and 4 edges

5264 is the smallest number so that it and its successor are both the product of 2 primes and the 4^th power of a prime

5265 is a Rhonda number

5269 is the number of binary rooted trees with 18 vertices

5271 is a value of n for which 2n and 7n together use each digit exactly once

5274 is the sum of its proper divisors that contain the digit 7

5278 is the number of ways, up to symmetry, to pick 3 elements of an 8×8 grid

5279 is the number permutations of {1,2,3,...,20} where adjacent numbers differ by no more than 2

5280 is the number of feet in a mile

5281 has a 4^th power that is the sum of four 4^th powers

5282 is the number of different arrangements (up to rotation and reflection) of 8 non-attacking rooks on a 8×8 chessboard

5284 and the two numbers before it and after it are all products of exactly 3 primes

5289 is a structured rhombic triacontahedral number

5291 is a value of n for which n(n+1) is a palindrome .

5292 = 28 + 0 + 0 + 5264 and has square 28005264

5293 is the smallest number that ends an arithmetic progression of 12 numbers with the same prime signature

5296 is the Entringer number E(8,3).

5306 is the smallest number whose 9^th power starts with 4 identical digits

5309 has the property that if each digit is replaced by its square , the resulting number is a square

5312 is the index of a prime Woodall number

5313 is the index of a triangular number containing only 3 different digits

5314 is a value of n for which |cos(n)| is smaller than any previous integer

5322 is the starting location of 7777 in the decimal expansion of π

5324 is the number of binary cube-free words of length 20

5327 is a value of n for which 2n and 7n together use each digit exactly once

5328 is the number of one-sided 6-knights

5332 is a Kaprekar constant in base 3

5335 is the magic constant of a 22×22 magic square

5336 is a house number

5340 is an octahedral number

5346 = 198 × 27 and each digit is contained in the equation exactly once

5349 = 12345 in base 8

5355 is an odd primitive abundant number (A091191, A006038)

5357 is the smallest number that can not be formed using the digit 1 at most 27 times, together with the symbols +, –, × and ÷

5358 are the first 8 digits of π ⁵³⁵⁸

5362 is the number of Chess positions that can be reached after 2 moves by white and 1 move by black

5364 is a value of n for which 3n and 7n together use each digit exactly once

5366 is the number of graphs with 8 vertices that have chromatic number 4

5367 uses the same digits as φ (5367)

5369 is a Wolstenholme number

5371 is a value of n for which n and 8n together use each digit 1-9 exactly once

5376 is the order of a perfect group

5382 is the number of non-intersecting rook paths joining opposite corners of a 6×4 chessboard

5383 is the number of triangles of any size contained in the triangle of side 27 on a triangular grid

5387 is the index of a prime Fibonacci number

5390 is the number of ways to 7-color the faces of a cube

5392 is a Leyland number

5399 has a cube whose digits occur with the same frequency

5400 is divisible by its reverse

5401 is a member of the Fibonacci -type sequence starting with 3 and 7

5405 is the smaller number in a Ruth-Aaron pair

5406 is the number of ways a 9×1 rectangle can be surrounded by 9×1 rectangles

5408 is an Achilles number

5409 and its reverse are both differences of positive cubes

5412 is a value of n so that n(n+4) is a palindrome

5414 is the number of binary partitions of 43

5418 is a value of n for which n and 7n together use each digit 1-9 exactly once

5419 is a Cuban prime

5422 is the number of semigroups of order 6 with 3 idempotents

5431 is the smallest number whose 4^th power contains 5 consecutive 9's

5432 has digits in arithmetic sequence

5434 is the sum of consecutive squares in 2 ways

5436 is the number of terms in the 10^th derivative of f(f(f(f(f(x)))))

5439 is a Rhonda number

5440 is the number of ways to legally add 2 sets of parentheses to a product of 15 variables

5443 is the smallest prime p with 17 consecutive quadratic residues mod p

5446 is the number of ways to to arrange the numbers 1-10 around a circle so that the sums of adjacent numbers are distinct

5448 is the number of ways to cut a 10×10 chessboard into 2 pieces with equal areas with a cut that only travels up and right

5455 is a Kaprekar number for cubes

5456 and its reverse are tetrahedral numbers

5457 is a number whose sum of divisors is a 5^th power

5460 is both a triangular number and 7 times a triangular number

5461 is a Poulet number

5462 is the number of ways to walk along 14 edges of a triangle and end at the original vertex

5463 has a 4^th power that is the sum of four 4^th powers

5464 is the number of subsets of {1,2,3,...,16} that have a sum divisible by 12

5469 has the property that e ^5469] is within .00003 of an integer

5471 contains no 0's in base 3 through base 10

5472 has a base 3 representation that ends with its base 4 representation

5473 has a base 3 representation that ends with its base 4 representation

5474 is a stella octangula number

5477 and its reverse are both one more than a square

5478 is the number of graphs with 10 vertices that have chromatic number 2

5479 is the number of bipartite graphs with 10 vertices

5482 is the number of 3×3 sliding puzzle positions that require exactly 16 moves to solve starting with the hole in the center

5483 is the number of unlabeled distributive lattices with 18 elements

5487 is the maximum number of pieces a torus can be cut into with 31 cuts

5488 is an Achilles number

5489 is the maximum number of regions a cube can be cut into with 32 cuts

5491 has a 4^th power that is the sum of four 4^th powers

5493 is the number of integers with complexity 30

5499 is the average of all the even 4-digit numbers

5504 is the number of series-parallel networks with 6 labeled edges

5505 is a value of n for which n!!! - 1 is prime

5507 has a square root whose decimal part starts with the digits 0-9 in some order

5508 is the generalized Catalan number C(13,5)

5509 is the number of multigraphs with 8 vertices and 9 edges

5513 is the number of self-avoiding walks of length 10

5525 is the smallest number that can be written as the sum of 2 squares in 6 ways

5530 is a hexagonal pyramidal number

5533 is the number of graphs with 10 vertices and 2 cycles

5536 is the 16^th Tetranacci number

5542 is the number of anisohedral 19-ominoes

5543 has a 4^th power that is the sum of four 4^th powers

5544 is the number of permutations of 9 items that fix 4 elements

5545 is a member of the Fibonacci -type sequence starting with 1 and 5

5551 is the number of trees on 17 vertices with diameter 6

5554 is a Kaprekar number for cubes

5555 is a repdigit

5557 contains no 0's in base 3 through base 10

5560 are the first 4 digits of 7⁵⁵⁶⁰

5561 has the property that the sum of its prime factors is equal to the product of its digits

5564 is an amicable number

5565 is a doubly triangular numbers

5566 is a pentagonal pyramidal number

5568 is the number of ways to put 8 checkers on an 8×8 checkerboard so that each row, column, and main diagonal contains exactly one checker

5571 is a perfect totient number

5573 is the number of digits in the 6^th Cullen prime

5576 is a decagonal pyramidal number

5585 is the number of monoids of order 7 with 2 idempotents

5586 does not occur in its factorial in base 2

5587 has a 5^th root that starts 5.61611166...

5588 is the index of a triangular number containing only 3 different digits

5591 is the smallest prime that is followed by 31 composite numbers .

5594 is the number of ways to dissect a 14-gon using non-crossing diagonals into polygons with an even number of sides.

5595 is the number of labeled mappings from 6 points to themselves with exactly 3 cycles.

5597 has a cube with only odd digits.

5600 is the number of self-complementary graphs with 13 vertices

5602 = 22222 in base 7 [A125725)

5604 is the number of partitions of 30

5610 is divisible by its reverse

5611 is the smallest number for which it and the 3 numbers before and after it all have φ (n) divisible by 10

5612 has the property that dropping its first and last digits gives its largest prime factor

5616 is the order of a non-cyclic simple group

5617 is a divisor of the sum of the 4^th powers of its divisors

5619 has a cube that contains the digits 5619 in reverse order

5620 is the smallest composite number which remains composite when preceded or followed by any digit

5623 and the primes preceding it and following it are all equal to 7 (mod 16)

5624 is the smallest number whose 4^th power has 15 digits

5625 has a cube that is the sum of 3 positive cubes

5629 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 16 stamps

5637 uses the same digits as φ (5637)

5638 is the number of 3×3 sliding puzzle positions that require exactly 17 moves to solve starting with the hole in a corner

5647 is the closest integer to 24^e

5651 is a number n for which n, n+2, n+6, and n+8 are all prime

5661 is the trinomial coefficient T(18,14)

5664 is a Rhonda number

5668 is the number of semigroups of order 6 with 5 idempotents

5669 is a value of n for which |cos(n)| is smaller than any previous integer

5670 is a value of n for which φ (n) and σ (n) are square

5671 is a triangular number that is the product of two primes

5673 is the smallest number whose 6^th power starts with 5 identical digits

5675 is the number of monic polynomials of degree 13 with integer coefficients whose complex roots are all in the unit disk

5678 has digits in arithmetic sequence

5679 is the number of drawings of the complete graph K₁₀ with a minimal number of crossings

5682 is the sum of its proper divisors that contain the digit 4

5689 is the largest 4-digit prime with strictly increasing digits

5691 is the number of different resistances that can be created in a circuit of 11 equal resistors

5692 is a number that does not have any digits in common with its cube

5693 = 5555 + 6 + 99 + 33

5694 = 17082 / 3, and each digit is contained in the equation exactly once

5696 is the smallest number whose square contains 4 consecutive 4's

5697 has a 21^st power that contains five 5's, six 6's, nine 9's, and seven 7's

5698 is the smallest number whose 8^th power starts with 5 identical digits

5700 is divisible by its reverse

5709 is a structured pentakis dodecahedral number

5711 is the smallest prime p with 18 consecutive quadratic residues mod p

5712 is the number of Gray codes for a 4-dimensional cube

5717 is a value of n for which the first n binary digits of π form a prime

5718 is the number of partitions of 54 into distinct parts

5719 is a Zeisel number

5720 is a dodecagonal pyramidal number

5721 is the number of graphs with 8 vertices that have chromatic number 3

5723 has the property that its square starts with its reverse

5729 has a 4^th power that is the sum of four 4^th powers

5731 is a value of n for which n (n+2) is a palindrome

5734 has a square that is a centered pentagonal number

5737 is the smallest number that can not be formed using the digit 1 at most 22 times, together with the symbols +, × and ^

5739 is a value of n for which 5n and 7n together use each digit exactly once

5740 = 7777 in base 9

5741 is the 11^th Pell number

5742 is a value of n for which 5n and 8n together use each digit exactly once

5751 is the number of ordered sequences of coins totaling 31 cents

5754 is the number of ways a loop can cross two parallel lines a total of 12 times

5755 is the sum of the digits of the 19^th Mersenne prime (A066538)

5760 is the order of a perfect group

5767 is the product of two consecutive primes

5768 is the 16^th tribonacci number

5770 is a value of n for which φ (n) and σ (n) are square

5772 are the first 4 decimal digits of Euler's constant

5773 is the index of a triangular number containing only 3 different digits

5774 is the smallest number whose square begins with four 3's

5775 is the smallest value of n for which both n and n+1 are abundant (A096399)

5776 is the square of the last half of its digits

5777 is the smallest multi-digit number which is not the sum of a prime and twice a square

5778 is the largest Lucas number which is also a triangular number

5781 is a centered tetrahedral number

5784 has a square whose digits each occur twice

5786 = 5555 + 77 + 88 + 66

5789 is a value of n for which n and 8n together use each digit 1-9 exactly once

5790 has the same digits as the 5790^th prime

5791 is a value of n for which n and 8n together use each digit 1-9 exactly once

5793 are the first 4 digits of 5793^e

5795 is a value of n such that the n^th Cullen number is prime

5796 = 138 × 42 and each digit is contained in the equation exactly once

5798 is the 11^th Motzkin number

5807 is the index of a Wagstaff prime

5813 is the concatenation of 3 consecutive Fibonacci numbers

5814 = ₁₉P ₃

5817 = 34902 / 6, and each digit is contained in the equation exactly once

5818 contains no 0's in base 3 through base 10

5819 has a sum of digits equal to its largest prime factor

5821 contains no 0's in base 3 through base 10

5822 is the number of conjugacy classes in the automorphism group of the 16 dimensional hypercube .

5823 is the smallest value of n for which n and 3n together use each digit 1-9 exactly once

5824 can be written as the difference between two positive cubes in more than one way

5825 are the first 4 digits of e⁵⁸²⁵

5830 is a weird number

5831 has a sum of digits equal to its largest prime factor

5832 is a value of n for which n and 3n together use each digit 1-9 exactly once

5834 is the number of digits of the 21^st perfect number [A061193)

5839 is a value of n for which n and 8n together use each digit 1-9 exactly once

5842 is a Padovan number

5843 has a 5^th root that starts 5.66666...

5844 is the number of ways to stack 34 boxes in a line so that each box lies on the table or on a box next to 2 boxes

5848 has a square that remains square when a 9 is appended to it

5850 is a value of n for which n-1 and n+1 are twin primes , and so are 2n-1 and 2n+1

5851 is a value of n for which n, n², and n³ have the same digit sum

5853 is the index of a triangular number containing only 3 different digits

5856 = 1 × 6 × 16 × 61

5858 has a square whose digits each occur twice

5859 can be written as the difference between two positive cubes in more than one way

5860 is the sum of the squares of 4 consecutive primes

5863 is the starting location of 7777 in the decimal expansion of π

5864 has a 14^th power that contains five 5's, eight 8's, six 6's, and four 4's

5865 is an enneagonal pyramidal number

5867 is a member of the Fibonacci -type sequence starting with 1 and 9

5868 is a value of n for which n, n², and n³ have the same digit sum

5870 has a digit sum smaller than its cube

5872 = 5555 + 88 + 7 + 222

5873 divides 1¹ + 2² + 3³ + ^{. . .} + 5873⁵⁸⁷³

5876 is the number of ways to color the vertices of a triangle with 26 colors, up to rotation

5877 is a value of n for which 5n and 8n, or 8n and 9n, together use each digit exactly once

5879 is the smallest number so that it and the next 10 numbers all have an odd number of prime factors

5880 is the Stirling number of the second kind S(10,7)

5885 is a number whose sum of divisors is a 5^th power

5886 is a value of n for which 3n and 5n together use each digit exactly once

5890 is a heptagonal pyramidal number

5892 is a value of n for which n and 8n together use each digit 1-9 exactly once

5895 is the number of necklaces possible with 7 beads, each being one of 5 colors

5896 is the number of ways to tile a 3×24 rectangle with 3×1 rectangles

5900 is the number of ways to place 32 points on a 16×16 grid so that no 3 points are on a line

5904 has a square comprised of the digits 1-8

5906 is the smallest number which is the sum of 2 rational 4^th powers but is not the sum of two integer 4^th powers

5909 is the number of symmetric plane partitions of 32

5913 = 1! + 2! + 3! + 4! + 5! + 6! + 7!

5914 = 0! + 1! + 2! + 3! + 4! + 5! + 6! + 7!

5915 is the sum of consecutive squares in 2 ways

5916 is a value of n for which n and 8n together use each digit 1-9 exactly once

5921 is a value of n for which n and 8n together use each digit 1-9 exactly once

5923 is the largest n so that Q (√n) has class number 7

5925 is the index of a triangular number containing only 3 different digits

5926 + φ (5926) = 8888

5929 is a square which is also the sum of 11 consecutive squares

5931 is the number of one-sided 7-kings

5934 is a value of n for which 5n and 7n together use each digit exactly once

5935 is a Smith brother

5936 is divisible by the digits it does not contain, and not divisible by the digits it contains

5938 is the number of binary partitions of 44

5939 is the smallest prime so that it and the next 2 primes are all equal to 3 (mod 7)

5940 is divisible by its reverse

5941 is the number of interior intersections when all the diagonals of a regular 22-gon are drawn

5943 is a value of n for which n, n+1, n+2, and n+3 have the same number of divisors

5950 is the sum of the digits of the 20^th Mersenne prime (A066538)

5953 and the primes preceding it and following it are all equal to 3 (mod 14)

5958 is the number of subsets of {1,2,3,...,16} that have a sum divisible by 11

5959 is the smaller number in a Ruth-Aaron pair

5960 is the larger number in a Ruth-Aaron pair

5963 is the number of intersections when all the diagonals of a regular 22-gon are drawn

5967 is a value of n for which 6n and 7n together use each digit exactly once

5968 has a square which uses the digits 0-7 each exactly once

5972 is the smallest number that appears in its factorial 8 times

5974 is the number of connected planar graphs with 8 vertices

5975 is a value of n for which σ (n) = σ (reverse(n))

5976 is a value of n for which n and 7n together use each digit 1-9 exactly once

5978 is a value of n where φ (n) is the product of the digits of n

5984 = ₃₄C ₃

5985 = ₂₁C ₄

5986 and its prime factors contain every digit from 1-9 exactly once

5993 is the largest number known which is not the sum of a prime and twice a square

5994 is the number of lattices on 10 unlabeled nodes

5995 is a palindromic triangular number

5996 is a truncated tetrahedral number

5999 is the smallest number whose digits add to 32