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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 9th 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 24th 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 8th root that starts 3.12345...
9070 has a 4th 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 23rd 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 10th 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 4th power that is the sum of four 4th powers
9139 = 39C 3
9152 = A068782(18) and its successor are both divisible by 4th 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 4th power, and a 5th 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 4th power that is the sum of four 4th 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 13th 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 8th power that contains the same digits as 35889
9349 is the 19th Lucas number
9350 appears inside its 4th 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 7th 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 15th power contains exactly the same digits as another 15th 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 29e
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 4th 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 5th 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 4th power that is the sum of four 4th 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 9551st 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 4th power that is the sum of four 4th 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 4th 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 4th 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 15th 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 4th power that is the sum of four 4th powers (A039664)
9878 has a 10th 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
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)