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Annotated version of "What's Special About This Number?" (Part 6)
From OeisWiki
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 6: The Numbers 6000 to 6999
6000 is the number of subsets of the 24^{th} roots of unity that add to 1
6001 has a cube that is a concatenation of other cubes
6002 is the number of digits of the 24^{th} Mersenne prime (A028335)
6003 has a square with the first 3 digits the same as the next 3 digits
6006 is the number of intersections when all the diagonals of a regular 21-gon are drawn
6008 = _{14}C _{6} + _{14}C _{0} + _{14}C _{0} + _{14}C _{8}
6009 is a strobogrammatic number
6011 is a member of the Fibonacci -type sequence starting with 3 and 8
6012 has a square with the last 3 digits the same as the 3 digits before that
6014 has a square that is formed by 3 squares that overlap by 1 digit
6016 is the maximum number of pieces a torus can be cut into with 32 cuts
6017 is a centered octahedral number
6018 is the maximum number of regions a cube can be cut into with 33 cuts
6020 is the number of Hamiltonian graphs with 8 vertices
6021 has a square that is formed by 3 squares that overlap by 1 digit
6024 is a value of n for which |cos(n)| is smaller than any previous integer
6025 are the last 4 digits of the sum of the first 6025 squares
6032 is the number of ways to place 2 non-attacking knights on a 9×9 chessboard
6035 is a number whose sum of divisors is a 5^{th} power
6040 is the number of ways to divide 6 couples into pairs where no pair is a couple
6048 is the order of a non-cyclic simple group
6050 has a sum of digits equal to its largest prime factor
6058 is a number that does not have any digits in common with its cube
6065 is the closest integer to 16^{π }
6070 is a structured truncated tetrahedral number
6072 is the order of a non-cyclic simple group
6073 is the order of a non-cyclic simple group
6075 is an Achilles number
6077 has a square with the last 3 digits the same as the 3 digits before that
6080 is the smallest number n>1 whose base 14 representation is equal to φ (n)
6081 has a cube that is the sum of 3 positive cubes
6083 has a square that is the sum of a cube and a 4^{th} power
6084 is the sum of the first 12 cubes
6092 is the number of 16-ominoes with a line of symmetry
6093 is a value of n for which 3n and 5n together use each digit exactly once
6095 is a rhombic dodecahedral number
6097 is an hexagonal prism number
6099 concatenated with its successor is square
6100 has the property that if each digit is replaced by its square , the resulting number is a square
6102 is the largest number n known where φ (n) is the reverse of n
6105 is a Huay rhombic dodecahedral number
6106 is a value of n for which 2φ (n) = φ (n+1)
6107 is a Perrin number
6111 is a value of n for which σ (n-1) = σ (n+1)
6119 is a strobogrammatic number
6120 is a highly abundant number (A002093)
6121 is the smallest number whose cube contains 4 consecutive 3's
6128 is a betrothed number
6137 is a centered dodecahedral number
6138 is the number of quasi-tetrominoes that fit inside a 7×7 grid
6141 is a Kaprekar constant in base 2
6142 is the number of inequivalent asymmetric Ferrers graphs with 34 points
6143 is the smallest prime that contains twelve 1's in binary
6144 = 16!!!!
6145 is a Friedman number
6155 is a member of the Fibonacci -type sequence starting with 2 and 5
6164 is the number of 11-ominoes that tile the plane using 180 degree rotations
6167 has a 4^{th} power that is the sum of four 4^{th} powers
6168 is the number of inequivalent Ferrers graphs with 34 points
6170 = 5 + 55 + 555 + 5555
6171 has the property that dropping its first and last digits gives its largest prime factor
6173 is a prime that remains prime if any digit is deleted
6174 is the Kaprekar constant for 4-digit numbers
6175 is the number of regions formed when all diagonals are drawn in a regular 21-gon
6176 is the last 4-digit sequence to appear in the decimal expansion of π
6179 is a value of n for which 4n and 5n together use each digit exactly once
6180 is the smallest number n with φ (n) = 2 reverse(n)
6181 is an octahedral number
6187 is a Smith brother
6188 = _{17}C _{5}
6189 is the number of ways to write 17 as an ordered sum of positive integers , where adjacent numbers are different
6194 is the number of ways to place a non-attacking white and black pawn on a 10×10 chessboard
6196 is the number of regions the complex plane is cut into by drawing lines between all pairs of 21^{st} roots of unity
6197 is a narcissistic number in base 6
6200 is a harmonic divisor number
6201 is the sum of the first 26 squares
6210 is the number of 5×5 matrices with non-negative entries with every row and column adding to 2
6211 is a Cuban prime
6216 has a square with the first 3 digits the same as the next 3 digits
6219 is a value of n for which 4n and 5n together use each digit exactly once
6220 = 44444 in base 6
6221 = 666 + 2222 + 2222 + 1111
6222 is the smallest number that can not be written as the sum of 2 triangular numbers and a power of 2
6223 = 666 + 2222 + 2 + 3333
6224 is the number of permutations of 8 elements have 4^{th} power equal to the identity element
6225 = 666 + 2 + 2 + 5555
6232 is an amicable number
6237 is a number whose sum of the squares of its divisors is a square
6239 , followed by 6239 7's, is prime
6240 is a highly abundant number (A002093)
6244 is a member of the Fibonacci -type sequence starting with 2 and 9
6245 is the smallest number whose square contains 4 consecutive internal 0's
6248 is the smallest number with the property that its first 8 multiples contain the digit 4
6249 is the smallest number with the property that its first 10 multiples contain the digit 4
6250 is a Leyland number
6256 is a hendecagonal pyramidal number
6257 is the number of essentially different ways to dissect a 20-gon into 9 quadrilaterals
6266 is a truncated octahedral number
6267 is the number of 15-iamonds with holes
6270 is a value of n for which n-1 and n+1 are twin primes , and so are 2n-1 and 2n+1
6271 is the smallest number requiring an addition chain of length 17
6272 is the number of ways to tile a 4×29 rectangle with 4×1 rectangles
6273 is the number of ways to 9-color the vertices of a pentagon, up to rotations and reflections
6274 has a cube whose digits occur with the same frequency
6276 is a value of n for which φ (n) = φ (reverse(n))
6279 is the number of subsequences of {1,2,3,...14} in which every odd number has an even neighbor
6280 has a sum of prime factors that is equal to the sum of the prime factors of the two preceding numbers
6290 is the number of 13-iamonds that do not tile the plane
6293 is the number of ordered partitions of 24 into distinct parts
6296 has a square with the first 3 digits the same as the next 3 digits
6297 is a value of n for which n and 5n together use each digit 1-9 exactly once
6299 is the smallest number with complexity 30
6300 is divisible by its reverse
6307 is the largest n so that Q (√n) has class number 8
6309 is the closest integer to 25^{e }
6310 is the smallest number whose 5^{th} power has 19 digits
6312 is the sum of its proper divisors that contain the digit 5
6318 is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .
6320 is the Entringer number E(8,4).
6322 is the number of idempotent functions from a set of 7 elements into itself.
6327 = 324 + 325 + . . . + 342 = 343 + 344 + . . . + 360
6331 has the same digits as the 6331^{st} prime
6336 is the number of ways to tile a 9×4 rectangle with 2×1 rectangles
6343 is the number of quasi-triominoes that fit inside a 14×14 grid
6347 has the same digits as the 6347^{th} prime
6348 is a pentagonal pyramidal number
6351 is the largest number known that is not the sum of 3 squares or cubes
6354 is the number of 14-iamonds that tile the plane
6360 is a value of n for which n-1 and n+1 are twin primes , and so are 3n-1 and 3n+1
6368 is an amicable number
6371 has a square that is the sum of 2 relatively prime cubes
6374 is a value of n for which 4n and 5n together use each digit exactly once
6375 has a square with the first 3 digits the same as the next 3 digits
6378 is the number of partitions of 55 into distinct parts
6379 is a value of n for which |cos(n)| is smaller than any previous integer
6380 is a value of n for which n! + 1 is prime
6381 is the smallest value of n for which n and 9n together use each digit 1-9 exactly once
6384 is an icosahedral number
6385 is the number of ways to stack 18 pennies in a line so that each penny lies on the table or on two pennies
6389 is the number of functional graphs on 11 vertices
6391 is a hexagonal pyramidal number
6395 is the number of ways to divide a 12×12 grid of points into two sets using a straight line
6396 is a divisor of the sum of the 4^{th} powers of its divisors
6397 has the same digits as the 6397^{th} prime
6399 and its successor are both divisible by 4^{th} powers
6400 is a square whose digits are non-increasing
6403 has a square with the first 3 digits the same as the last 3 digits
6404 is a value of n for which n!! - 1 is prime
6406 is the number of permutations of 8 elements where every cycle has equal length
6408 is the sum of the squares . of the first 13 primes
6409 is a house number
6411 is a truncated cube number
6424 is the number of minimal covers of a set containing 6 elements
6427 is the number of ways a 6×6 square can be tiled with 1×1 and 2×2 squares
6432 has the same digits as the 6432^{nd} prime
6434 is the number of divisors of the 18^{th} perfect number
6435 = _{15}C _{7}
6443 has a cube whose digits occur with the same frequency
6444 is the smallest number whose 5^{th} power starts with 5 identical digits
6445 , followed by 6445 1's, is prime
6454 is the smallest value of n for which π(10n) = n
6455 is the smallest value of n for which the n^{th} prime begins with the digits of n
6456 is a value of n for which the n^{th} prime begins with the digits of n
6457 is a value of n for which the n^{th} prime begins with the digits of n
6458 would be prime if preceded and followed by a 1, 3, 7, or 9
6459 is a value of n for which the n^{th} prime begins with the digits of n
6460 is a value of n for which the n^{th} prime begins with the digits of n
6462 divides the sum of the digits of 6462!
6466 is the largest known value of n for which the n^{th} prime begins with the digits of n
6471 is a value of n for which n and 9n together use each digit 1-9 exactly once
6472 is the number of polyominoes with 9 or fewer squares
6475 is a value of n for which π(n) is the product of the digits of n
6479 is a value of n for which n and 8n together use each digit 1-9 exactly once
6481 = (3^{12} + 1) / (3^{4} + 1)
6487 is the number of partitions of 51 in which no part occurs only once
6488 would be prime if preceded and followed by a 1, 3, 7, or 9
6489 is half again as large as the sum of its proper divisors
6490 is the number of ways to place 2 non-attacking bishops on a 11×11 chessboard
6498 is the index of a triangular number containing only 3 different digits
6500 is a number n whose sum of the factorials of its digits is equal to π(n)
6501 has a square whose reverse is also a square
6505 is the number of 9-hexes without holes
6506 is a value of n for which the first n binary digits of π form a prime
6510 is a number n whose sum of the factorials of its digits is equal to π(n)
6511 is a number n whose sum of the factorials of its digits is equal to π(n)
6514 is the sum of the 4^{th} powers of the digits of the sum of the 4^{th} powers of the digits of itself
6517 has a sum of digits equal to its largest prime factor
6521 is a number n whose sum of the factorials of its digits is equal to π(n)
6523 is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .
6524 has the property that its square starts with its reverse
6525 is a centered icosahedral number
6526 is the smallest number whose 10^{th} power contains exactly the same digits as another 10^{th} power
6527 is a value of n for which φ (n) = φ (n-1) + φ (n-2)
6529 is a Proth prime
6532 is a member of the Fibonacci -type sequence starting with 1 and 6
6533 is the number of digits of the 25^{th} Mersenne prime (A028335)
6534 is a value of n for which 3n and 7n together use each digit exactly once
6537 is the smallest value of n for which the numbers n-6 through n+6 can not be written as the sum of 2 squares
6540 is the number of terms in the 17^{th} derivative of f(f(f(x)))
6543 has a square root that has four 8's immediately after the decimal point
6544 is a number n whose 9^{th} root has a decimal part that begins with the digits of n
6545 and its reverse are tetrahedral numbers
6547 is the number of binary 4×4 matrices with no row or column containing 3 consecutive 1's
6552 is the number of different full houses in 5 card poker with one joker
6553 is a Lucas 5-step number
6556 is the largest palindrome that can be made using 5 digits and the 4 arithmetic operations
6557 is the product of two consecutive primes
6560 is the smallest number n where n and n+1 are both products of 7 or more primes
6561 = 3^{8}
6569 is a value of n for which one less than the product of the first n primes is prime
6572 is the number of 9-hexes
6576 = (6! - 6) + (5! - 5) + (7! - 7) + (6! - 6)
6578 is the smallest number which can be written as the sum of three 4^{th} powers in 2 ways
6579 is the number of ways to color the vertices of a triangle with 27 colors, up to rotation
6580 is the maximum number of regions a cube can be cut into with 34 cuts
6581 has the same digits as the 6581^{st} prime
6583 is a value of n for which σ (φ (n)) = 2σ (n)
6588 is the number of sided 12-iamonds
6593 = 6 + 5555 + 999 + 33
6594 is a value of n for which 5n and 7n together use each digit exactly once
6596 has a square comprised of the digits 0-7
6601 is a Carmichael number
6603 is a number whose square and cube use different digits
6608 is the maximum number of regions space can be divided into by 28 spheres
6609 has a 4^{th} power that is the sum of four 4^{th} powers
6611 is a value of n such that the n^{th} Cullen number is prime
6615 is an odd abundant number (A005101, A005231)
6620 is the number of 11-ominoes that tile the plane
6623 has the property that the sum of its prime factors is equal to the product of its digits
6630 is the number of triangles of any size contained in the triangle of side 29 on a triangular grid
6636 has exactly the same digits in 3 different bases
6639 divides 1^{1} + 2^{2} + 3^{3} + ^{ . . .} + 6639^{6639}
6642 can be written as the sum of 2 or 4 positive 4^{th} powers
6643 is the smallest number which is palindromic in bases 2 and 3
6647 has a sum of digits equal to its largest prime factor
6651 is the index of a triangular number containing only 3 different digits
6653 , when concatenated with 4 less than itself, is square
6654 is the smallest number whose decimal part of its 4^{th} root starts with the digits 0-9 in some order
6663 is a value of n for which σ (n) is a repdigit
6665 is a centered tetrahedral number
6666 is a repdigit
6667 is the number of self-dual planar graphs with 24 edges
6668 is the number of trees on 21 vertices with diameter 5
6669 is the sum of 3 consecutive cubes
6680 = 6666 + 6 + 8 + 0
6681 = 6666 + 6 + 8 + 1
6682 = 6666 + 6 + 8 + 2
6683 = 6666 + 6 + 8 + 3
6684 = 6666 + 6 + 8 + 4
6685 = 6666 + 6 + 8 + 5
6686 = 6666 + 6 + 8 + 6
6687 = 6666 + 6 + 8 + 7
6688 = 6666 + 6 + 8 + 8
6689 = 6666 + 6 + 8 + 9
6694 is a value of n for which the sum of the first n primes is square
6699 is a strobogrammatic number
6700 has a cube that contains the digits 6700 in reverse order
6704 is the number of rooted 8-hexes
6706 is the number of Hamiltonian paths in a 8×5 rectangle graph
6712 is the index of a triangular number containing only 3 different digits
6714 is the index of a triangular number containing only 3 different digits
6716 is the 4-digit string that appears latest in the decimal expansion of π
6720 = _{8}P _{5}
6721 is a composite value of n that divides the (n-1)^{st} Fibonacci number
6723 is a value of n for which 3n and 8n together use each digit exactly once
6726 is the 10^{th} Pell-Lucas number
6728 is the number of domino tilings of a 6×6 square
6729 is the smallest value of n for which n and 2n together use each digit 1-9 exactly once
6731 would be prime if preceded and followed by a 1, 3, 7, or 9
6732 is a value of n for which _{2n}C _{n} is divisible by n^{2}
6734 is a value of n for which |cos(n)| is smaller than any previous integer
6735 is a stella octangula number
6736 is the number of 3×3 sliding puzzle positions that require exactly 17 moves to solve starting with the hole in the center
6740 is the number of 13-iamonds that do not tile the plane
6741 is a value of n for which n and 8n together use each digit 1-9 exactly once
6742 has a square where the first 6 digits alternate
6743 is the number of binary 4×5 matrices with no consecutive 1's in any row or column
6745 is the maximum value of n so that there exist 4 denominations of stamps so that every postage from 1 to n can be paid for with at most 25 stamps
6751 is the number of digits of the 23^{rd} perfect number (A061193)
6754 is the smallest number in base 9 to have 5 different digits
6756 has a cube that is the sum of 3 positive cubes
6759 is the number of graphs with 10 vertices and 11 edges
6764 is the sum of the first 18 Fibonacci numbers
6765 is the 20^{th} Fibonacci number
6768 has a 9^{th} root that starts 2.664444666...
6769 is the Stirling number of the first kind s(8,4)
6772 has a square whose digits each occur twice
6779 = 6666 + 7 + 7 + 99
6780 has the same digits as the 6780^{th} prime
6786 is a triangular number whose internal digits are triangular and whose external digits are triangular
6788 is the smallest number with multiplicative persistence 6
6789 is the largest 4-digit number with increasing digits
6791 is a value of n for which n and 8n together use each digit 1-9 exactly once
6792 is a value of n for which n and 2n together use each digit 1-9 exactly once
6793 is the smallest prime so that it and the next 2 primes all end in 3
6794 has the property that dropping its first and last digits gives its largest prime factor
6797 is a number whose sum of divisors is a 5^{th} power
6801 has a 4^{th} power that is the sum of four 4^{th} powers
6802 is the number of ways to move a rook from corner to opposite corner on a 6×6 chessboard
6811 is not the sum of a square , a cube , a 4^{th} power, and a 5^{th} power
6813 is the smallest number whose 6^{th} power has 24 digits
6816 is the index of a triangular number containing only 3 different digits
6818 = 1^{8} + 2^{8} + 3^{8}
6819 = 20457 / 3, and each digit is contained in the equation exactly once
6820 is the number of regions formed when all diagonals are drawn in a regular 23-gon
6822 uses the same digits as φ (6822)
6825 is an odd primitive abundant number (A091191, A006038)
6828 is the number of ways to start with a knight in the corner of an 8×8 chessboard, make 8 moves, and end on the same square
6831 is a structured truncated octahedral number
6837 is the number of 8-digit squares
6839 is a value of n for which n and 8n together use each digit 1-9 exactly once
6840 is the number of ways to place 2 non-attacking kings on a 11×11 chessboard
6842 is the number of partitions of 31
6845 would be prime if preceded and followed by a 1, 3, 7, or 9
6849 is a value of n for which 2n and 3n together use each digit exactly once
6850 is the smallest value of n for which n, n+1, n+2, n+3, n+4, and n+5 have the same number of prime factors
6853 is a value of n for which n, n+1, n+2, and n+3 have the same number of divisors
6859 = 19^{3}
6860 is a heptagonal pyramidal number
6861 is a value of n for which σ (n-1) + σ (n+1) = σ (2n)
6863 is a prime that is the sum of the square of a prime and the cube of a prime
6864 = 6666 + 88 + 66 + 44
6865 is the maximum value of n so that there exist 5 denominations of stamps so that every postage from 1 to n can be paid for with at most 17 stamps
6867 can be written as the sum of 2, 3, 4, or 5 positive cubes
6868 is the larger number in a Ruth-Aaron pair
6874 is equal to the sum of its anti-divisors
6875 is 3-automorphic
6879 is the number of planar partitions of 15
6880 is a vampire number
6886 is a palindrome in base 9 and in base 10
6888 has a square with 3/4 of the digits are the same
6889 is a strobogrammatic square
6895 is a value of n for which 2n and 7n together use each digit exactly once
6896 has a square root whose decimal part starts with the digits 0-9 in some order
6900 is the number of ways to place 2 non-attacking knights on a 11×11 chessboard
6902 is the number of Hamiltonian paths of a 3×10 rectangle graph
6903 is a value of n for which σ (n-1) = σ (n+1)
6905 has a 5^{th} root whose decimal part starts with the digits 1-9 in some order
6912 = 6 × 9 × 1 × 2^{7}
6917 is a value of n for which n! - 1 is prime
6918 = 20754 / 3, and each digit is contained in the equation exactly once
6919 is the number of non-invertible knots with 13 crossings
6922 is the number of polycubes containing 8 cubes
6924 is the magic constant of a 24×24 magic square
6926 has a square whose digits each occur twice
6927 is a value of n for which n and 2n together use each digit 1-9 exactly once
6928 is the number of inequivalent binary linear codes of length 11
6930 is the square root of a triangular number
6931 has the same digits as the 6931^{st} prime
6935 is the smallest number whose cube contains six 3's
6936 is the number of ways to legally add 2 sets of parentheses to a product of 16 variables
6939 is a value of n for which 3n and 5n together use each digit exactly once
6940 is the sum of its proper divisors that contain the digit 3
6941 has a square whose digits each occur twice
6942 is the number of labeled topologies with 5 elements
6944 is the number of degree sequences for graphs with 6 vertices
6949 is the smallest number that can not be written as the sum of 3 volumes of rectangular boxes with integer dimensions less than 16
6951 has exactly the same digits in 3 different bases
6952 = 1738 × 4 and each digit from 1-9 is contained in the equation exactly once
6953 = 66 + 999 + 5555 + 333
6954 is the trinomial coefficient T(19,15)
6956 is the number of triangles formed by drawing all diagonals of a regular 12-gon
6960 is the number of ways to place 2 non-attacking queens on a 10×10 chessboard
6966 is the number of planar graphs with 8 vertices
6969 is a strobogrammatic number
6972 is the number of possible positions in Checkers containing 2 checkers
6976 is the number of binary 5×5 matrices A with the property that A^{2}=0 (mod 2)
6982 is a value of n for which the sum of the first n composite number numbers is a square
6983 is the smallest prime that can only be made into 1 other prime by changing a single digit
6984 can be written as the sum of 2, 3, 4, or 5 positive cubes
6985 is the smallest number that can be written as the sum of 3 or more consecutive squares , or as the sum of 3 or more consecutive cubes
6987 is the number of digits of the 26^{th} Mersenne prime (A028335)
6989 has the property that the concatenation of its prime factors in increasing order is a square
6991 is a value of n for which reverse(φ (n)) = φ (reverse(n))
6996 is a palindrome n so that n(n+8) is also palindromic
6998 is a member of the Fibonacci -type sequence starting with 4 and 9
6999 is the smallest number whose digits add to 33