Annotated version of "What's Special About This Number?" (Part 7)

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 7: The Numbers 7000 to 7999

7000 has a sum of digits equal to its largest prime factor (A052021)

7001 is the number of 13-hexes that tile the plane by translation

7002 is the number of arrangements of 4 non-attacking queens on a 8×8 chessboard

7003 is the number of graphs with 9 vertices that have 8 automorphisms

7014 has a square with the last 3 digits the same as the 3 digits before that

7015 has a cube root whose decimal part starts with the digits 1-9 in some order

7019 is a prime that remains prime if any digit is deleted (A051362)

7028 is the smallest multi-digit number n, when written in base 17, gives a divisor of n

7030 is an octagonal pyramidal number (A002414)

7032 is the number of ternary square-free words of length 24

7039 = 28156 / 4, and each digit is contained in the equation exactly once

7040 has a sum of digits equal to its largest prime factor

7055 is a Lucas-Carmichael number (A006972)

7056 is a square that is the product of two triangular numbers (A169835, A169836)

7057 is a Cuban prime (A007645)

7060 has the property that the sum of the squares of its divisors ends with the digits 7060

7066 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 13 stamps (A001211)

7068 is the number of series-reduced planted trees with 11 leaves (A000669)

7071 is the smallest number whose square contains 4 consecutive 9's

7072 is the generalized Catalan number C(10,7)

7073 is a Leyland number

7075 is the number of ways to stack 35 boxes in a line so that each box lies on the table or on a box next to 2 boxes

7084 is the generalized Catalan number C(19,4)

7089 is a value of n for which |cos(n)| is smaller than any previous integer

7092 is the number of possible positions in Othello after 3 moves by both players

7093 has a 6^th root that starts 4.38333833...

7094 is the number of ways to place 34 points on a 17×17 grid so that no 3 points are on a line

7096 is the number of 8-digit perfect powers

7098 is the trinomial coefficient T(14,9) (A000574)

7101 has a 4^th power that is the sum of four 4^th powers

7102 is the index of a Fibonacci number whose first 9 digits are the digits 1-9 rearranged

7106 is an octahedral number

7107 has a square whose digits each occur twice

7108 is the number of partitions of 56 into distinct parts

7117 is a number whose sum of divisors is a 5^th power

7119 has the same digits as the 7119^th prime

7120 is the number of 2×2 singular matrices mod 10

7122 = 7² + 8³ + 9⁴

7123 is the number of 2-connected graphs with 8 vertices

7140 is the largest number which is both triangular and tetrahedral

7142 is the smallest number that can not be written as the sum of 2 volumes of rectangular boxes with integer dimensions less than 20

7143 is 7-automorphic

7145 has a square with the first 3 digits the same as the next 3 digits

7150 has a sum of digits equal to its largest prime factor

7152 has a square with the first 3 digits the same as the next 3 digits

7159 has a square with the first 3 digits the same as the next 3 digits

7161 is a Kaprekar constant in base 2

7164 is a value of n for which n⁸, n⁹, n¹⁰, and n¹¹ have the same digit sum

7170 is a value of n for which σ (n-1) = σ (n+1)

7172 is a Kaprekar number for cubes

7174 is the maximum number of pieces a torus can be cut into with 34 cuts

7175 is a centered octahedral number

7176 is the maximum number of regions a cube can be cut into with 35 cuts

7187 is the smallest number that can not be formed using the digits 0-8 at most once, together with the symbols +, –, × and ÷

7188 is the number of ways to permute 5 red, 5 white, and 5 blue balls

7189 is the number of ways to color the vertices of a square with 13 colors, up to rotation

7192 is a weird number

7193 is a right-truncatable prime

7197 is the smallest number whose 7^th power has 27 digits

7200 is the order of a perfect group

7201 is the number of 2×2 singular matrices mod 19

7209 has a 4^th power that is the sum of four 4^th powers

7212 is the number of unordered ways to write 1 as a sum of reciprocals of integers no larger than 20

7225 is the number of ways to 17-color the faces of a tetrahedron

7226 has a cube root that starts 19.3330030330...

7230 is the sum of consecutive squares in 2 ways

7235 is a value of n for which 4n and 5n together use each digit exactly once

7236 uses the same digits as φ (7236)

7240 = 1111 in base 19

7241 is the number of asymmetric trees with 19 vertices

7245 appears inside its 4^th power

7248 is the number of lines through exactly 2 points of a 14×14 grid of points

7253 has a square that remains square when a 6 is appended to it

7254 = 186 × 39 and each digit is contained in the equation exactly once

7256 is a value of n for which n, n+1, n+2, and n+3 have the same number of divisors

7260 is a doubly triangular numbers

7269 is a value of n for which n and 2n together use each digit 1-9 exactly once

7271 and its reverse are both differences of positive cubes

7272 is a Kaprekar number (A006886)

7281 is a value of n for which 3n and 7n together use each digit exactly once

7285 has a 7^th power that contains the same digits as 544¹⁰

7286 is the number of subsets of {1,2,3,...,16} that have a sum divisible by 9

7293 is a value of n for which n and 2n together use each digit 1-9 exactly once

7295 is a value of n for which 4n and 5n together use each digit exactly once

7297 is a Proth prime

7306 is the smallest number whose 7^th power starts with 7 identical digits

7311 is the number of symmetric plane partitions of 33

7312 is a value of n for which n and 8n together use each digit 1-9 exactly once

7314 is the smallest number so that it and its successor are both products of 4 distinct primes

7315 = ₂₂C ₄

7318 is the number of functions from 10 unlabeled points to themselves

7320 is the number of triangles of any size contained in the triangle of side 30 on a triangular grid

7321 is the number of intersections when all the diagonals of a regular 24-gon are drawn

7322 is the number of 3×3 sliding puzzle positions that require exactly 17 moves to solve starting with the hole on a side

7326 = 1 × 22 × 333

7327 is a number whose sum of divisors is a 5^th power

7329 is a value of n for which n and 2n together use each digit 1-9 exactly once

7330 is the number of unsymmetrical ways to dissect a regular 14-gon into 12 triangles

7331 is a right-truncatable prime

7333 is a right-truncatable prime

7336 is the number of ways to color the vertices of a triangle with 28 colors, up to rotation

7337 is a hexagonal pyramidal number

7338 is the closest integer to 17^π

7339 has a 4^th power that is the sum of four 4^th powers

7341 has the same digits as the 7341^st prime

7342 is the number of ways to stack 29 pennies in contiguous rows so that each penny lies on the table or on two pennies

7344 is a value of n for which 4n and 7n together use each digit exactly once

7345 has the same digits as the 7345^th prime

7351 is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

7353 is the largest number n known so that both n and n³ have only odd digits.

7356 is a value of n for which 5n and 7n together use each digit exactly once

7358 is a composite number that remains composite when preceded or followed by any digit

7359 is a Lucas 6-step number

7360 can be written as the product of a number and its reverse in 2 different ways

7361 is the number of ways to play the first 5 moves in Checkers

7364 is a value of n for which n and 8n together use each digit 1-9 exactly once

7366 is the maximum number of regions space can be divided into by 29 spheres

7371 has a base 2 representation that begins with its base 9 representation

7375 is a member of the Fibonacci -type sequence starting with 1 and 4

7376 is a structured truncated tetrahedral number

7380 is the number of numbers with 4 or fewer digits that do not contain any 0's

7381 = 11111 in base 9

7383 has a 4^th power that is 1/2 of the sum of three 4^th powers

7384 has the same digits as the 7384^th prime

7385 is a Keith number

7387 is the product of two consecutive primes

7393 is a right-truncatable prime

7396 has a 4^th root whose decimal part starts with the digits 1-9 in some order

7403 is the smallest number that can not be formed using the digit 1 at most 28 times, together with the symbols +, –, × and ÷

7404 = 6 + 66 + 666 + 6666

7410 = 361 + 362 + . . . + 380 = 381 + 382 + . . . + 399

7413 is the number of even permutations on 8 elements with no fixed points

7414 is a value of n for which φ (n) = φ (reverse(n))

7416 is a value of n for which n and 8n together use each digit 1-9 exactly once

7420 is the number of permutations of 8 items that fix 2 elements

7421 is a value of n for which 4n and 5n together use each digit exactly once

7422 is the sum of its proper divisors that contain the digit 7

7424 and its successor are both abundant (A096399)

7425 is an odd primitive abundant number (A091191, A006038)

7427 is the number of inequivalent asymmetric Ferrers graphs with 35 points

7429 is the product of 3 consecutive primes

7430 is the number of labeled commutative monoids of order 5

7433 is a prime that remains prime if any digit is deleted

7435 is a cubic star number

7436 is the number of 6×6 alternating sign matrices

7444 is a value of n for which |cos(n)| is smaller than any previous integer

7447 is a palindrome in base 2 and in base 10

7448 is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

7456 is the number of inequivalent Ferrers graphs with 35 points

7462 is the number of multigraphs with 26 vertices and 4 edges

7464 is a structured hexagonal diamond number

7465 = 54321 in base 6

7469 is the smallest number that can not be written as the sum of 2 volumes of rectangular boxes with integer dimensions less than 21

7471 is a centered cube number

7473 is a Tribonacci -like number starting from 1, 1, and 1

7475 has a sum of digits equal to its largest prime factor

7480 is a value of n for which _2nC _n is divisible by n²

7485 is the number of conjugacy classes of the alternating group A₃₅

7488 = (12 × 13 × 14 × 15 × 16) / (12 + 13 + 14 + 15 + 16)

7490 has a square with the last 3 digits the same as the 3 digits before that

7491 has a base 8 representation which is the reverse of its base 7 representation

7494 is the sum of its proper divisors that contain the digit 4

7496 = 777 + 44 + 9 + 6666

7497 is a hendecagonal pyramidal number

7499 is the smallest number whose 8^th power has 31 digits

7500 is the order of a perfect group

7508 would be prime if preceded and followed by a 1, 3, 7, or 9

7509 has a 6^th root whose decimal part starts with the digits 1-9 in some order

7512 is the sum of its proper divisors that contain the digit 5

7515 has the property that the sum of its prime factors is equal to the product of its digits

7519 is a member of the Fibonacci -type sequence starting with 1 and 7

7524 is the number of rectangles with corners on an 12×12 grid of points

7525 has a square with the last 3 digits the same as the 3 digits before that

7528 is the number of ways, up to rotation and reflection, of dissecting a regular 14-gon into 12 triangles

7531 has digits in arithmetic sequence

7532 has a square comprised of the digits 0-7

7535 has a square whose digits each occur twice

7541 is an Eisenstein-Mersenne prime (A066408)

7542 is a value of n for which 4n and 7n together use each digit exactly once

7546 is the number of series-reduced planted trees with 19 vertices

7547 is the maximum number of regions a circle can be cut into by joining 21 points on the circumference with straight lines

7549 is the largest known prime p where no numbers of the form p-n² are prime

7551 is a value of n for which φ (n) + φ (n+1) divides σ (n) + σ (n+1)

7552 is the number of arrangements of 6 non-attacking queens on a 10×6 chessboard

7557 is a palindrome that is the sum of the first 37 palindromes

7560 is the smallest number with 64 divisors

7561 is a Markov number

7562 would be prime if preceded and followed by a 1, 3, 7, or 9

7574 is the sum of consecutive squares in 2 ways

7581 is the number of monotone Boolean functions of 5 variables

7586 = 777 + 55 + 88 + 6666

7588 is the smallest multiple of 28 whose digits add to 28

7590 is a number whose sum of divisors is a 4^th power

7595 is the number of simplicial polyhedra with 12 vertices

7597 is a number whose sum of divisors is a 5^th power

7600 is a substring of any power of itself

7614 is a value of n for which n and 7n together use each digit 1-9 exactly once

7615 is a value of n for which σ (n+1) = 2σ (n)

7617 is a Hexanacci number

7618 has a cube that contains only digits 4 and smaller.

7620 is the number of multigraphs with 5 vertices and 14 edges

7625 is a value of n for which σ (φ (n)) = 2σ (n)

7627 is a value of n for which σ (φ (n)) = 2σ (n)

7629 is a value of n for which n and 5n together use each digit 1-9 exactly once

7632 is a value of n for which 5n and 6n together use each digit exactly once

7635 is a centered tetrahedral number

7639 is the number of rooted ternary trees with 13 vertices

7647 is a Keith number

7648 is the number of ways a 10×1 rectangle can be surrounded by 10×1 rectangles

7650 can be written as the product of a number and its reverse in 2 different ways

7651 is a value of n for which _2nC _n is not divisible by 3, 5, or 7

7652 is a value of n for which n² and n³ use the same digits

7654 has digits in arithmetic sequence

7658 is the largest number known that does not have any digits in common with its cube

7659 is the number of planar graphs with 22 vertices, all with degree 5 or more

7663 is the product of two primes which are reverses of each other

7664 is the Entringer number E(8,6).

7665 is a Kaprekar constant in base 2

7667 is a palindrome in base 6 and in base 10

7669 is the number of integers with complexity 31

7672 = 777 + 6666 + 7 + 222

7673 is the smallest number with the property that its first 8 multiples contain the digit 3

7679 = 7 + 6666 + 7 + 999

7680 is the number of possible rook moves on a 16×16 chessboard

7681 is a Proth prime

7683 is a truncated tetrahedral number

7685 is the number of necklaces possible with 18 beads, each being one of 2 colors

7686 is a value of n for which 7n and 9n together use each digit exactly once

7688 is an Achilles number

7692 is a value of n for which n and 2n together use each digit 1-9 exactly once

7693 is a value of n for which the sum of the first n primes is a palindrome

7695 and its successor are both divisible by 4^th powers

7698 has a square with the first 3 digits the same as the next 3 digits

7700 is a value of n for which 2φ (n) = φ (n+1)

7703 has a 4^th power that is the sum of four 4^th powers

7710 is the number of degree 17 irreducible polynomials over GF(2)

7712 is the number of necklaces (that can't be turned over) possible with 17 beads, each being one of 2 colors

7713 is a value of n for which 4n and 9n together use each digit exactly once

7714 is the sum of the first 28 squares

7721 is the smallest value of n for which 3ⁿ contains 8 consecutive 3's

7724 is the smallest number that can not be written using +, ×, and 5 Fibonacci numbers

7727 is the index of a Fibonacci number whose first 9 digits are the digits 1-9 rearranged

7732 and the two numbers before it and after it are all products of exactly 3 primes

7734 is the sum of its proper divisors that contain the digit 8

7736 is the number of labeled Eulerian digraphs with 5 vertices

7738 has the property that dropping its first and last digits gives its largest prime factor

7739 is a Padovan number

7741 is the number of trees with 15 vertices

7743 is the smallest number whose 9^th power has 35 digits

7744 is the smallest known square with no isolated digits

7745 and its reverse are both one more than a square

7746 is the number permutations of {1,2,3,...,21} where adjacent numbers differ by no more than 2

7752 is the generalized Catalan number C(14,5)

7754 is the number of binary cube-free words of length 21

7755 is the index of a prime Woodall number

7765 is the number of ways to tile a 7×5 rectangle with integer -sided squares

7770 = ₃₇C ₃

7772 has a square root whose decimal part starts with the digits 1-9 in some order

7775 = 55555 in base 6

7776 is a 5^th power whose digits are non-increasing

7777 is a Kaprekar number (A006886)

7778 is the closest integer to 27^e

7785 is a value of n for which 5n and 6n together use each digit exactly once

7788 is the index of a triangular number containing only 3 different digits

7792 has a square that is the sum of a cube and 5^th power

7793 is the smallest prime so that it and the next 5 primes are all equal to 5 (mod 6)

7795 has the same digits as the 7795^th prime

7799 is a value of n for which |cos(n)| is smaller than any previous integer

7800 is the order of a non-cyclic simple group

7803 is an Achilles number

7805 is the maximum number of pieces a torus can be cut into with 35 cuts

7807 is the maximum number of regions a cube can be cut into with 36 cuts

7808 is the number of 4×4 sliding puzzle positions that require exactly 12 moves to solve starting with the hole in a corner

7810 has the property that its square is the concatenation of two consecutive numbers

7811 is the number of ordered sequences of coins totaling 32 cents

7812 = 222222 in base 5

7820 is the Stirling number of the second kind S(17,15)

7821 is a value of n for which 2n and 9n together use each digit exactly once

7824 is a value of n for which 5n and 7n together use each digit exactly once

7825 is a rhombic dodecahedral number

7826 is the number of necklaces possible with 6 beads, each being one of 6 colors

7827 has a square whose digits each occur twice

7835 would be prime if preceded and followed by a 1, 3, 7, or 9

7846 is a factor of 7847784878497850

7848 is the number of connected 5-regular graphs with 12 vertices

7849 is the number of connected 6-regular graphs with 12 vertices

7851 = 7777 + 8 + 55 + 11

7852 = 1963 × 4, and each digit from 1-9 is contained in the equation exactly once

7853 is the largest prime factor of 11! - 1

7854 is a number whose sum of divisors is a 4^th power

7856 and its successor are both the product of a prime and the 4^th power of a prime

7860 is the number of nonisomorphic 3-state automata with binary inputs and outputs

7874 is the smallest number n for which n concatenated with n+2 is a square

7875 is an odd abundant number (A005101, A005231)

7880 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 26 stamps

7882 is a structured pentagonal hexacontahedral number

7884 is a value of n for which 2n and 5n together use each digit exactly once

7887 is the index of a pentagonal number which is twice another pentagonal number

7888 is a value of n where φ (n) is the product of the digits of n

7890 is an icosahedral number

7894 is a value of n for which n and 8n together use each digit 1-9 exactly once

7895 is the number of multigraphs with 6 vertices and 11 edges

7905 is a Kaprekar constant in base 2

7908 has the same digits as the 7908^th prime

7909 is a Keith number

7912 is a weird number

7913 is a value of n for which σ (n-1) = σ (n+1)

7917 is the number of partitions of 57 into distinct parts

7919 is the 1000^th prime

7920 is the order of the smallest sporadic group

7921 is the square of a Fibonacci number

7922 has the property that the sum of its prime factors is equal to the product of its digits

7923 is a value of n for which n and 2n together use each digit 1-9 exactly once

7926 is the diameter of the earth in miles

7928 is a Friedman number

7931 is a heptagonal pyramidal number

7932 is a value of n for which n and 2n together use each digit 1-9 exactly once

7936 is the 5^th tangent number

7937 is the smallest number whose cube contains 5 consecutive 9's

7939 , when followed by any of its digits, is prime

7941 = 7777 + 9 + 44 + 111

7942 = 7777 + 99 + 44 + 22

7946 = 7777 + 99 + 4 + 66

7953 is the number of domino tilings of a 3×14 rectangle

7954 is the smallest value of n for which 5ⁿ + n is prime

7956 is a value of n for which n and 4n together use each digit 1-9 exactly once

7957 is a Poulet number

7960 is a structured deltoidal hexacontahedral number

7964 is a value of n for which φ (n) = φ (reverse(n))

7969 has a square that is formed by 3 squares that overlap by 1 digit

7980 is the smallest number whose divisors contain every digit at least 7 times

7983 is a Lucas 8-step number

7986 = 11 × 22 × 33

7992 can be written as the difference between two positive cubes in more than one way

7993 is one less than twice its reverse

7994 has a 5^th power that is closer to a cube than a square

7997 is a palindrome in base 4 and in base 10

7999 , when followed by any of its digits, is prime