Annotated version of "What's Special About This Number?" (Part 9)

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 9: The Numbers 9000 to 9999

9000 is the index of a triangular number containing only 3 different digits

9002 is a value of n such that n(n+7) is a palindrome

9005 is the number of inequivalent Ferrers graphs with 36 points

9006 is a strobogrammatic number

9009 is a centered cube number

9011 has a square that is the concatenation of two consecutive odd numbers

9012 is the sum of its proper divisors that contain the digit 5

9016 is the number of perfect squared rectangles of order 16

9018 has a square with the last 3 digits the same as the 3 digits before that

9020 is the number of ways to color the vertices of a triangle with 30 colors, up to rotation

9023 has the property that the concatenation of its prime factors in increasing order is a square

9024 is the number of regions formed when all diagonals are drawn in a regular 24-gon

9025 is a Friedman number

9028 is the number of ways to tile a 9×4 rectangle with integer -sided squares

9032 would be prime if preceded and followed by a 1, 3, 7, or 9

9036 has a 9^th power that contains the same digits as 3585¹⁰

9037 is a value of n for which 2n and 7n together use each digit exactly once

9038 is the number of conjugacy classes of the alternating group A₃₆

9042 is the trinomial coefficient T(11,4)

9045 is the number of ways to 18-color the faces of a tetrahedron

9048 is the number of regions the complex plane is cut into by drawing lines between all pairs of 24^th roots of unity

9049 is an Eisenstein-Mersenne prime (A066408, A066408)

9052 is the maximum number of regions space can be divided into by 31 spheres

9055 is the index of a triangular number containing only 3 different digits

9056 is a cubic star number

9059 has an 8^th root that starts 3.12345...

9070 has a 4^th root whose decimal part starts with the digits 1-9 in some order

9072 has a base 2 and base 3 representation that end with its base 6 representation

9073 has a base 2 and base 3 representation that end with its base 6 representation

9074 has a base 3 representation that ends with its base 6 representation

9077 is a Markov number

9078 has a cube whose digits occur with the same frequency

9079 has a square that is the concatenation of two consecutive decreasing numbers

9086 is the number of regions formed when all diagonals are drawn in a regular 23-gon

9091 is the unique prime whose reciprocal has period 10

9093 has a square with the first 3 digits the same as the next 3 digits

9099 is the number of ways to 3-color the faces of a dodecahedron

9101 has a square where the first 6 digits alternate

9104 has a square with the first 3 digits the same as the next 3 digits

9105 is the number of possible positions in Checkers after 6 moves

9108 is a heptagonal pyramidal number

9109 is the number of regions the complex plane is cut into by drawing lines between all pairs of 23^rd roots of unity

9113 is a narcissistic number in base 5

9115 has a base 3 representation that begins with its base 6 representation

9116 is a strobogrammatic number

9117 is a value of n for which 6n and 7n together use each digit exactly once

9119 is the number of symmetric plane partitions of 34

9121 is the number of possibilities for the last 5 digits of a square

9126 is a pentagonal pyramidal number

9134 has a 10^th root whose decimal part starts with the digits 1-9 in some order

9135 is a value of n for which 2n and 7n together use each digit exactly once

9137 has a 4^th power that is the sum of four 4^th powers

9139 = ₃₉C ₃

9152 = A068782(18) and its successor are both divisible by 4^th powers

9153 is a value of n for which 2n and 3n together use each digit exactly once

9154 is a value of n for which φ (n) and σ (n) are square

9156 is a value of n for which n and 8n together use each digit 1-9 exactly once

9158 is a value of n for which n and 8n together use each digit 1-9 exactly once

9162 is a value of n for which 5n and 8n together use each digit exactly once

9168 = 27504 / 3, and each digit is contained in the equation exactly once

9172 is the number of connected planar maps with 7 edges

9174 is the sum of its proper divisors that contain the digit 5

9176 is the maximum number of pieces a torus can be cut into with 37 cuts

9178 is the maximum number of regions a cube can be cut into with 38 cuts

9179 is a value of n for which φ (n) = φ (n-1) + φ (n-2)

9182 is a value of n for which 4n and 5n together use each digit exactly once

9183 is the number of sets of distinct positive integers with mean 8

9185 is a value of n for which 2n and 7n together use each digit exactly once

9189 is the number of sided 10-ominoes

9191 is not the sum of a square , a cube , a 4^th power, and a 5^th power

9196 has the property that dropping its first and last digits gives its largest prime factor

9198 is the number of ternary square-free words of length 25

9201 is a truncated octahedral number

9214 = A001524(30) is the number of ways to stack 30 pennies in contiguous rows so that each penny lies on the table or on two pennies

9216 is a Friedman number

9217 is the total number of digits of all binary numbers of length 1-10

9219 is a value of n for which |cos(n)| is smaller than any previous integer

9224 is an octahedral number

9233 is the number of different arrangements (up to rotation and reflection) of 13 non-attacking queens on a 13×13 chessboard

9234 is the number of multigraphs with 7 vertices and 10 edges

9235 is the number of 13-iamonds

9237 is a value of n for which n and 5n together use each digit 1-9 exactly once

9240 = ₂₂P ₃

9241 is a Cuban prime

9243 has a 4^th power that is the sum of four 4^th powers

9248 is the number of possible rook moves on a 17×17 chessboard

9250 = (10³ + 10⁴ + 10⁵ + 10⁶) / (3 × 4 × 5 × 6)

9251 has a square whose digits each occur twice

9252 is the number of necklaces with 10 white and 10 black beads

9253 is the smallest number that appears in its factorial 9 times

9261 is a Friedman number

9267 is a value of n for which n and 2n together use each digit 1-9 exactly once

9268 is a value of n for which 2φ (n) = φ (n+1)

9272 is a weird number

9273 is a value of n for which n and 2n together use each digit 1-9 exactly once

9282 is the product of three consecutive Fibonacci numbers

9284 is the number of ways to place 2 non-attacking bishops on a 12×12 chessboard

9285 is the number of 16-hexes with reflectional symmetry

9286 is a narcissistic number in base 7

9287 is the number of stretched 10-ominoes

9288 can be written as the sum of 2, 3, 4, or 5 positive cubes

9289 is a Tetranacci -like number starting from 1, 1, 1, and 1

9298 has the property that the concatenation of its prime factors in increasing order is a square

9304 = 65128 / 7, and each digit is contained in the equation exactly once

9305 has the property that if each digit is replaced by its square , the resulting number is a square

9306 is a value of n for which 3n and 5n together use each digit exactly once

9310 is a decagonal pyramidal number

9311 is the index of a prime Fibonacci number

9313 , when followed by any of its digits, is prime

9314 is the 13^th Iccanobif number

9315 is a value of n for which 2n and 3n together use each digit exactly once

9316 is a value of n for which n and 8n together use each digit 1-9 exactly once

9321 is a value of n for which n and 8n together use each digit 1-9 exactly once

9324 is the reciprocal of the sum of the reciprocals of 14652 and its reverse

9327 is a value of n for which n and 2n together use each digit 1-9 exactly once

9330 is the Stirling number of the second kind S(10,3)

9331 has the property that the sum of its prime factors is equal to the product of its digits

9339 is a value of n for which φ (n) = φ (n-2) - φ (n-1)

9347 is a value of n for which the sum of square -free divisors of n and n+1 are the same

9348 has a 8^th power that contains the same digits as 3588⁹

9349 is the 19^th Lucas number

9350 appears inside its 4^th power

9352 is a value of n for which n and 8n together use each digit 1-9 exactly once

9360 is a value of n for which σ (n-1) = σ (n+1)

9362 = 22222 in base 8

9363 is the number of tilted rectangles with vertices in a 15×15 grid

9364 is the number of connected digraphs with 5 vertices

9367 is a value of n for which n, n+1, n+2, and n+3 have the same number of divisors

9371 is a prime that remains prime when preceded and followed by one, two, three, or four 3's

9374 is a value of n for which φ (σ (n)) = φ (n)

9375 has a cube that ends with those digits

9376 is an automorphic number

9377 is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

9378 is a value of n for which 4n and 5n together use each digit exactly once

9380 is the number of lines through exactly 2 points of a 15×15 grid of points

9382 is a value of n for which 4n and 5n together use each digit exactly once

9383 is the index of a Fibonacci number whose first 9 digits are the digits 1-9 rearranged

9385 is the sum of consecutive squares in 2 ways

9386 = 99 + 333 + 8888 + 66

9387 is a Smith brother

9391 has a square with the first 3 digits the same as the last 3 digits

9393 is the number of non-isomorphic 3×3×3 Rubik's cube positions that require exactly 5 quarter turns to solve

9394 is a value of n so that n(n+8) is a palindrome

9396 is the number of symmetric 3×3 matrices in base 6 with determinant 0

9403 = 65821 / 7, and each digit is contained in the equation exactly once

9406 is the index of a triangular number containing only 3 different digits

9407 has a 7^th root whose decimal part starts with the digits 1-9 in some order

9408 is the number of reduced 6×6 Latin squares (A000315)

9413 has a cube whose digits occur with the same frequency

9415 is the sum of the first 19 numbers that have digit sum 19

9416 is a value of n for which n and 8n together use each digit 1-9 exactly once

9421 is a value of n for which n and 8n together use each digit 1-9 exactly once

9424 has the property that the fractional part of π ⁹⁴²⁴ begins .9424...

9426 is a value of n for which 5n and 7n together use each digit exactly once

9427 is the smallest number that can not be formed using the digit 1 at most 29 times, together with the symbols +, –, × and ÷

9428 is the smallest number whose square begins with four 8's

9431 is a number n for which n, n+2, n+6, and n+8 are all prime

9432 is the number of 3-colored rooted trees with 6 vertices

9436 is the smallest number whose 15^th power contains exactly the same digits as another 15^th power

9439 is prime , and 5 closest primes are all smaller

9444 has a square with the first 3 digits the same as the next 3 digits

9445 is the closest integer to 29^e

9450 is the denominator of ζ (8) / π ⁸

9451 is the number of binary rooted trees with 19 vertices

9452 is the smallest number whose cube contains 5 consecutive 4's

9455 is the sum of the first 30 squares

9465 is an hexagonal prism number

9468 is the sum of its proper divisors that contain the digit 7

9471 is an octagonal pyramidal number (A002414)

9473 is a Proth prime

9474 is a narcissistic number

9477 is the maximum determinant of a binary 13×13 matrix

9481 is a number whose sum of divisors is a 4^th power

9489 is the closest integer to π ⁸

9493 is a member of the Fibonacci -type sequence starting with 1 and 9

9496 is the number of 10×10 symmetric permutation matrices

9497 is the number of bicentered trees with 16 vertices

9499 has a 5^th power whose first few digits are 77337377...

9500 is a hexagonal pyramidal number

9504 is a betrothed number

9513 is the smallest number without increasing digits that is divisible by the number formed by writing its digits in increasing order

9519 has a 4^th power that is the sum of four 4^th powers

9520 is an enneagonal pyramidal number

9523 is a value of n for which 4n and 5n together use each digit exactly once

9529 is the number of 3×3 sliding puzzle positions that require exactly 18 moves to solve starting with the hole in a corner

9531 is the index of a prime Woodall number

9538 is a value of n for which 4n and 5n together use each digit exactly once

9541 is a value of n for which n and 8n together use each digit 1-9 exactly once

9542 is the number of ways to place a non-attacking white and black pawn on a 11×11 chessboard

9551 has the same digits as the 9551^st prime

9552 and the following 34 numbers are composite

9555 is an odd primitive abundant number (A091191, A006038)

9563 = 9 + 5555 + 666 + 3333

9564 is the number of paraffins with 10 carbon atoms

9568 = 9 + 5 + 666 + 8888

9574 is a value of n for which |cos(n)| is smaller than any previous integer

9576 = 19!!!!!

9583 is the number of subsets of {1, 2, 3, ... 20} that do not contain solutions to x + y = z

9592 is the number of primes with 5 or fewer digits

9596 is the index of a triangular number containing only 3 different digits

9601 is a Proth prime

9602 has the property that if each digit is replaced by its square, the resulting number is a square

9605 , when concatenated with 4 less than itself, is square

9608 is the number of digraphs with 5 vertices

9615 is the smallest number whose cube starts with 5 identical digits

9616 is an icosahedral number

9623 is the number of symmetric 10-cubes

9625 has a square formed by inserting a block of digits inside itself

9627 is a value of n for which n and 5n together use each digit 1-9 exactly once

9629 is a value of n for which 2n and 7n together use each digit exactly once

9632 is the number of different arrangements of 4 non-attacking queens on a 4×14 chessboard

9633 is a Smith brother

9634 is a Smith brother

9639 has a 4^th power that is the sum of four 4^th powers

9643 is the smallest number that can not be formed using the numbers 2⁰, 2¹, ... , 2⁷, together with the symbols +, –, × and ÷

9648 is a factor of the sum of the digits of 9648⁹⁶⁴⁸

9653 = 99 + 666 + 5555 + 3333

9658 = 99 + 666 + 5 + 8888

9660 is a truncated tetrahedral number

9670 is the number of 8-digit triangular numbers

9673 is the number of triangles of any size contained in the triangle of side 33 on a triangular grid

9677 is a prime that remains prime if any digit is deleted

9682 is a value of n for which n!! - 1 is prime

9689 is the exponent of a Mersenne prime (A000043, A000668)

9691 has the property that the concatenation of its prime factors in increasing order is a square

9695 is the sum of the digits of 5⁵⁵

9696 is a strobogrammatic number

9700 is the number of inequivalent 4-digit strings, where two strings are equivalent if turning one upside down gives the other

9701 has a square whose digits each occur twice

9707 does not occur in its factorial in base 2

9709 has a cube whose digits occur with the same frequency

9711 uses the same digits as π (9711)

9716 is the number of Pyramorphix puzzle positions that require exactly 5 moves to solve

9720 is the order of a perfect group

9721 is the largest prime factor of 1234567

9723 is a value of n for which n and 5n together use each digit 1-9 exactly once

9724 = 1111 in base 21

9726 is the smallest number in base 5 whose square contains the same digits in the same proportion

9728 can be written as the sum of 2, 3, 4, or 5 positive cubes

9738 is the number of trees on 22 vertices with diameter 5 (A000147)

9747 is an Achilles number (A052486)

9748 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 14 stamps (A001211)

9751 is the number of possible configurations of pegs (up to symmetry) after 8 jumps in solitaire (A112737)

9753 is a value of n for which 4n and 5n together use each digit exactly once

9754 is the number of paths between opposite corners of a 3×5 rectangle graph (A013992)

9760 can be written as the product of a number and its reverse in 2 different ways (A066531)

9764 would be prime if preceded and followed by a 1, 3, 7, or 9 (A059677)

9765 is an odd primitive abundant number (A091191, A006038)

9767 is the largest 4 digit prime composed of concatenating two 2 digit primes (A168499)

9768 = 2 × 22 × 222 (A084034)

9770 is the number of Hamiltonian cycles of a 4×12 rectangle graph (A006864)

9775 is a number n so that the sum of the digits of nⁿ-1 is divisible by n (A109675)

9777 is the number of graphs on 8 vertices with no isolated vertices (A006651)

9779 has a square root that has four 8's immediately after the decimal point

9784 is the number of 2 state Turing machines which halt (A004147)

9786 has a square whose digits each occur twice (A052049)

9789 is the smallest number that appears in its factorial 11 times (A061014)

9790 is the number of ways to place 2 non-attacking kings on a 12×12 chessboard (A061995)

9792 is the number of partitions of 59 into distinct parts (A078408)

9793 is the smallest number that can be written as the sum of 4 distinct positive cubes in 5 ways (A025421)

9796 has the property that dropping its first and last digits gives its largest prime factor (A114565)

9797 is the product of two consecutive primes (A006094)

9798 is a number whose sum of divisors is a 4^th power (A019422)

9799 is a number whose sum of squares of the divisors is a square (A046655)

9800 is the largest 4-digit number with single digit prime factors (A085868)

9801 is 9 times its reverse (A031877)

9802 , when concatenated with one less than it, is square (A054214)

9805 is the number of subsequences of {1,2,3,...15} in which every odd number has an even neighbor (A007483)

9809 is a stella octangula number (A007588)

9823 is the number of centered trees with 16 vertices (A000676)

9828 is the order of a non-cyclic simple group (A001034)

9831 has a base 6 representation which is the reverse of its base 5 representation

9839 would be prime if preceded and followed by a 1, 3, 7, or 9 (A059677, A059694)

9841 = 111111111 in base 3 (A125118, A055129, A003462)

9843 is the number of vertices in a Sierpinski triangle of order 8 (A067771)

9849 is a centered tetrahedral number (A005894)

9854 is the index of a triangular number containing only 3 different digits (A119207)

9855 is a rhombic dodecahedral number (A005917)

9856 is the number of ways to place 2 non-attacking knights on a 12×12 chessboard (A172132)

9857 is a Proth prime (A080076)

9858 is a number whose sum of divisors is a 4^th power (A019422)

9861 is a dodecagonal pyramidal number (A007587)

9862 is the number of knight's tours on a 6×6 chessboard (A001230)

9865 is the number of digits in the 15^th Fermat number (A057755)

9868 is the number of hydrocarbons with 10 carbon atoms (A002986)

9871 is the largest 4-digit prime with different digits (A007810)

9872 = 8 + 88 + 888 + 8888 (A099675)

9876 is the largest 4-digit number with different digits

9877 has a 4^th power that is the sum of four 4^th powers (A039664)

9878 has a 10^th power whose first few digits are 88448448...

9880 = ₄₀C ₃ (A010990, A004337)

9886 is a strobogrammatic number (A000787)

9888 is the number of connected graphs with 8 vertices whose complements are also connected (A054915)

9894 is the number of 3-colored trees with 7 vertices (A038060)

9896 is the number of Pyraminx puzzle positions that require exactly 6 moves to solve (A079744)

9900 = 10011010101100₂ = 9900₁₀ = 1881₁₉ = 1199₂₁, each using two digits the same number of times

9901 is the only prime known whose reciprocal has period 12 (A046107, A007615)

9910 is the number of fixed 9-ominoes (A001168)

9911 has the property that the sum of its prime factors is equal to the product of its digits (A067173, A065774)

9912 is the number of graceful permutations of length 14 (A006967)

9913 , when followed by any of its digits, is prime (A007811)

9918 is the maximum number of pieces a torus can be cut into with 38 cuts (A003600)

9919 can be written as the difference between two positive cubes in more than one way (A034179, A038864)

9920 is the maximum number of regions a cube can be cut into with 39 cuts (A000125)

9928 is a value of n for which reverse(φ (n)) = φ (reverse(n)) (A069282)

9929 is the number of 3×3 sliding puzzle positions that require exactly 26 moves to solve starting with the hole on a side (A089483)

9933 = 441 + 442 + . . . + 462 = 463 + 464 + . . . + 483 (A059270)

9941 is the exponent of a Mersenne prime (A000043, A000668)

9944 = 10011011011000₂ = 9944₁₀ = 2E2E₁₅ = 11BB₂₁, each using two digits the same number of times

9945 = 17!!!! (A081407)

9951 is the number of ways to color the vertices of a triangle with 31 colors, up to rotation (A006527)

9959 is a member of the Fibonacci -type sequence starting with 2 and 5 (A001060)

9960 is the number of 3×3×3 sliding puzzle positions that require exactly 8 moves to solve (A090573)

9966 is the largest 4-digit strobogrammatic number (A000787)

9973 is the largest 4-digit prime (A003618)

9976 has a square formed by inserting a block of digits inside itself (A045953, A046838)

9984 is the maximum number of regions space can be divided into by 32 spheres (A046127)

9985 is the number of hyperbolic knots with 13 crossings (A052408)

9988 is the number of prime knots with 13 crossings (A002863)

9992 is the number of 2×2×2 Rubik's cube positions that require exactly 5 moves to solve (A079761)

9995 has a square formed by inserting a block of digits inside itself (A045953, A046838)

9996 has a square formed by inserting a block of digits inside itself (A045953, A046838)

9998 is the smallest number n for which the concatenation of n, (n+1), ... (n+21) is prime (A052079)

9999 is a Kaprekar number (A006886) (A006886)