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# Annotated version of "What's Special About This Number?" (Part 3)

## 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 3: The Numbers 3000 to 3999

**3000** is the number of symmetric arrangements of 7 non-attacking queens on a 7×7 chessboard

**3001** is 1/24 of the 24^{th} Fibonacci number (A000045)

**3003** is the only number known to appear 8 times in Pascal's triangle

**3006** has a square with the last 3 digits the same as the 3 digits before that

**3008** is the number of symmetric plane partitions of 29

**3010** is the number of partitions of 27 (A000041)

**3012** is the sum of its proper divisors that contain the digit 5

**3015** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**3016** is a value of n for which n φ (n) is a palindrome

**3020** is the closest integer to π ^{7}

**3024** = _{9}P _{4}

**3025** is the sum of the first 10 cubes

**3028** are the first 4 digits of 5^{3028}

**3031** is the number of 7-kings

**3032** is the number of trees on 19 vertices with diameter 5

**3036** is the sum of its proper divisors that contain the digit 5

**3038** has a square that remains square when a 9 is appended to it

**3045** = 196 + 197 + . . . + 210 = 211 + 212 + . . . + 224

**3049** is the number of ways to tile a 8×4 rectangle with integer -sided squares

**3053** in hexadecimal spells the word BED

**3057** is the number of rooted ternary trees with 12 vertices

**3058** is the number of 7-digit triangular numbers

**3059** is a centered cube number

**3060** = _{18}C _{4}

**3063** is a perfect totient number

**3068** is the number of 10-ominoes that tile the plane

**3069** is a Kaprekar constant in base 2

**3070** is the number of paraffins with 9 carbon atoms

**3074** is the number of binary partitions of 37

**3077** is the rectilinear crossing number of complete graph K_{23}

**3078** is a pentagonal pyramidal number

**3080** is the number of drawings of the complete graph K_{9} with a minimal number of [http://3084 is the number of 3×3 sliding puzzle positions that require exactly 15 moves to solve starting with the hole in the center

**3081** is a doubly triangular numbers

**3087** is an Achilles number

**3089** is the smallest prime so that it and the next 2 primes all end in 9

**3092** is a structured truncated tetrahedral number

**3094** = 21658 / 7, and each digit is contained in the equation exactly once

**3096** is the number of 3×3×3 sliding puzzle positions that require exactly 7 moves to solve

**3097** is the largest known number n with the property that in every base, there exists a number that is n times the sum of its digits

**3101** is the number of ways to color the vertices of a triangle with 21 colors, up to rotation

**3103** = _{22}C _{3} + _{22}C _{1} + _{22}C _{0} + _{22}C _{3}

**3105** is a member of the Fibonacci -type sequence starting with 2 and 7

**3106** is both the sum of the digits of the 16^{th} and the 17^{th} Mersenne prime (A066538)

**3107** is the number of ways to divide a 10×10 grid of points into two sets using a straight line

**3109** is the smallest prime n so that n/π(n) > 7

**3110** = 22222 in base 6

**3114** has a square containing only 2 digits

**3115** has the property that if each digit is replaced by its square , the resulting number is a cube

**3119** is a right-truncatable prime

**3120** is the product of the first 6 Fibonacci numbers

**3121** = 3121_{5} + 3121_{7} + 3121_{8}

**3122** is the number of ordered sequences of coins totaling 29 cents

**3124** = 44444 in base 5

**3125** is a strong Friedman number

**3126** is a Sierpinski Number of the First Kind

**3127** is the product of two consecutive primes

**3135** is the smallest order of a cyclotomic polynomial whose factorization contains 7 as a coefficient

**3136** is a square that remains square if all its digits are decremented

**3137** is the number of planar partitions of 17

**3139** is the 9^{th} central trinomial coefficient

**3141** is the integer part of 1000 π

**3146** is a structured deltoidal hexacontahedral number

**3148** has a square with the first 3 digits the same as the next 3 digits

**3150** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**3156** is the sum of its proper divisors that contain the digit 5

**3159** is the number of trees with 14 vertices

**3160** is the largest known value of n for which _{2n}C _{n} is not divisible by the first 5 primes

**3161** is the smallest number whose square begins with three 9's

**3163** is the smallest number whose square has 7 digits

**3168** has a square whose reverse is also a square

**3169** is a Cuban prime

**3171** is the sum of the squares of 3 consecutive primes

**3174** is the first of four consecutive squareful numbers

**3178** = 4321 in base 9

**3179** is the number of 13-ominoes that tile the plane by translation

**3180** has a base 3 representation that ends with its base 5 representation

**3181** has a base 3 representation that ends with its base 5 representation

**3182** has a base 3 representation that ends with its base 5 representation

**3184** is a value of n for which |cos(n)| is smaller than any previous integer

**3185** is the number of ways to legally add 2 sets of parentheses to a product of 13 variables

**3186** is a value of n for which _{2n}C _{n} is not divisible by 3, 5, or 7

**3187** is the smallest value of n for which n and 8n together use each digit 1-9 exactly once

**3190** is a narcissistic number in base 7

**3191** is the smallest number whose reciprocal has period 29

**3192** is the number of planar graphs with 8 vertices, all with degree 2 or more

**3200** is the number of graceful permutations of length 13

**3203** has the property that if each digit is replaced by its square , the resulting number is a square

**3210** is the smallest 4-digit number with decreasing digits

**3212** = 3^{7} + 2^{9} + 1^{7} + 2^{9}

**3214** is the maximum number of regions a circle can be cut into by joining 17 points on the circumference with straight lines

**3216** is the smallest number with the property that its first 6 multiples contain the digit 6

**3217** is the exponent of a Mersenne prime (A000043, A000668)

**3218** has the property that the concatenation of its prime factors in increasing order is a square

**3225** is the number of symmetric 3×3 matrices in base 5 with determinant 0

**3226** is the number of 12-iamonds without holes

**3229** is a value of n for which one more than the product of the first n primes is prime

**3232** is the number of isomers of C_{12}H_{24} without any double bonds

**3240** is the number of 3×3×3 Rubik's cube positions that require exactly 3 moves to solve

**3242** has a square with the first 3 digits the same as the next 3 digits

**3243** in hexadecimal spells the word CAB

**3244** is the number of asymmetric trees with 18 vertices

**3245** in hexadecimal spells the word CAD

**3248** is the number of legal bishop moves in Chess

**3249** is the smallest square that is comprised of two squares that overlap in one digit

**3250** is a value of n for which _{2n}C _{n} is not divisible by 3, 5, or 7

**3251** is a number n for which n, n+2, n+6, and n+8 are all prime

**3252** is the number of graphs with 9 vertices and 11 edges

**3254** = 33 + 2222 + 555 + 444

**3259** = 33 + 2222 + 5 + 999

**3262** is the number of graphs with 9 vertices that have 6 automorphisms

**3264** is the number of partitions of 49 into distinct parts

**3267** = 12345 in base 7

**3274** = 303022_{4} = 101044_{5}, each using 3 different digits exactly twice

**3276** = _{28}C _{3}

**3277** is a Poulet number

**3280** = 11111111 in base 3

**3281** is the sum of consecutive squares in 2 ways

**3282** is the sum of its proper divisors that contain the digit 4

**3283** is the number of 3×3 sliding puzzle positions that require exactly 15 moves to solve starting with the hole on a side

**3290** is an enneagonal pyramidal number

**3292** is the number of ways to tile a 4×27 rectangle with 4×1 rectangles

**3294** is a value of n for which 6n and 7n together use each digit exactly once

**3297** is a value of n for which 5n and 7n together use each digit exactly once

**3300** is the number of non-isomorphic " groupoids on 4 elements

**3301** is a value of n for which the n^{th} Fibonacci number begins with the digits in n

**3302** is the maximum number of pieces a torus can be cut into with 26 cuts

**3303** is a centered octahedral number

**3304** is the maximum number of regions a cube can be cut into with 27 cuts

**3305** is the number of rectangles with corners on an 10×10 grid of points

**3311** is the sum of the first 21 squares

**3312** = 33^{2} + 12^{2}

**3313** is the smallest prime number where every digit d occurs d times

**3318** has exactly the same digits in 3 different bases

**3320** has a base 4 representation that ends with 3320

**3321** has a base 4 representation that ends with 3321

**3322** has a base 4 representation that ends with 3322

**3323** has a base 4 representation that ends with 3323

**3324** 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 20 stamps

**3325** is the smallest number that can not be written as the sum of 3 volumes of rectangular boxes with integer dimensions less than 13

**3326** is the smallest integer ratio of a 17-digit number to its product of digits

**3329** is a Padovan number

**3330** is a value of n for which n-1 and n+1 are twin primes , and so are 2n-1 and 2n+1

**3331** is the number of monoids of order 7 with 3 idempotents

**3333** is a repdigit

**3334** is the number of 12-iamonds

**3335** is the smallest number whose square contains 4 consecutive 2's

**3337** has a cube with only odd digits.

**3338** is a member of the Fibonacci -type sequence starting with 3 and 7

**3340** = 3333 + 3 + 4 + 0

**3341** = 3333 + 3 + 4 + 1

**3342** = 3333 + 3 + 4 + 2

**3343** = 3333 + 3 + 4 + 3

**3344** = 3333 + 3 + 4 + 4

**3345** = 3333 + 3 + 4 + 5

**3346** = 3333 + 3 + 4 + 6

**3347** = 3333 + 3 + 4 + 7

**3348** = 3333 + 3 + 4 + 8

**3349** = 3333 + 3 + 4 + 9

**3358** is the sum of the squares . of the first 11 primes

**3360** = _{16}P _{3}

**3361** is the number of quasi-triominoes that fit inside a 12×12 grid

**3362** has a square whose digits each occur twice

**3363** is a number n for which n^{2}+1 is double another square

**3366** = (1^{9} + 2^{9} + 3^{9}) / (1 × 2 × 3)

**3367** is the smallest number which can be written as the difference of 2 cubes in 3 ways

**3368** is the number of ways that 5 non-attacking bishops can be placed on a 5×5 chessboard

**3369** is a Kaprekar constant in base 4

**3375** is a Friedman number

**3376** is the number of digits of the 23^{rd} Mersenne prime (A028335)

**3378** is a Friedman number

**3379** is a number whose square and cube use different digits

**3380** would be prime if preceded and followed by a 1, 3, 7, or 9

**3381** is the number of ways to 14-color the faces of a tetrahedron

**3382** is a value of n for which 2φ (n) = φ (n+1)

**3383** has the property that the sum of its prime factors is equal to the product of its digits

**3386** has a square whose digits each occur twice

**3390** is a value of n for which n-1 and n+1 are twin primes , and so are 2n-1 and 2n+1

**3400** is a truncated tetrahedral number

**3402** can be written as the sum of 2, 3, 4, or 5 positive cubes

**3403** is a triangular number that is the product of two primes

**3404** is the number of binary partitions of 38

**3405** is a structured great rhombicosidodecahedral number

**3408** = 3^{3} + 4^{4} + 5^{5}

**3410** is a truncated square pyramid number

**3411** is the number of inequivalent asymmetric Ferrers graphs with 31 points

**3412** = 2^{2} + 3^{3} + 4^{4} + 5^{5}

**3413** = 1^{1} + 2^{2} + 3^{3} + 4^{4} + 5^{5}

**3417** is a hexagonal pyramidal number

**3420** is the order of a non-cyclic simple group

**3427** is a member of the Fibonacci -type sequence starting with 1 and 5

**3431** is the number of inequivalent Ferrers graphs with 31 points

**3432** is the 7^{th} central binomial coefficient

**3433** is a narcissistic number in base 6

**3435** = 3^{3} + 4^{4} + 3^{3} + 5^{5}

**3439** is a rhombic dodecahedral number

**3440** is the closest integer to 20^{e }

**3444** is a stella octangula number

**3447** is the smallest value of n for which 2n and 5n together use the digits 1-9 exactly once

**3451** is the number of conjugacy classes of the alternating group A_{31}

**3456** has digits in arithmetic sequence

**3457** is a Proth prime

**3459** has a 6^{th} root that starts 3.88888...

**3461** is a number n for which n, n+2, n+6, and n+8 are all prime

**3465** = 15!!!!

**3468** = 68^{2} - 34^{2}

**3476** is a value of n for which n!! - 1 is prime

**3478** has the property that dropping its first and last digits gives its largest prime factor

**3480** is a Perrin number

**3486** has a square that is formed by 3 squares that overlap by 1 digit

**3487** is the number of squares in a 14×14 grid of squares with diagonals drawn

**3488** has a 5^{th} root that starts 5.11111...

**3489** is the smallest number whose square has the first 3 digits the same as the last 3 digits

**3492** is the number of labeled semigroups of order 4

**3498** is a number whose sum of divisors is a 5^{th} power

**3499** in hexadecimal spells the word DAB

**3501** is a value of n for which n, 2n, 3n, and 4n all use the same number of digits in Roman numerals .

**3507** is a value of n for which n! - 1 is prime

**3510** = 6666 in base 8

**3511** is the largest known Wieferich prime

**3521** = 3333 + 55 + 22 + 111

**3522** is the sum of its proper divisors that contain the digit 7

**3525** is a Pentanacci number

**3527** is the number of ways to fold a strip of 10 stamps

**3528** is an Achilles number

**3531** is a value of n for which φ (n) = φ (n-2) - φ (n-1)

**3536** is a heptagonal pyramidal number

**3539** is a value of n for which |cos(n)| is smaller than any previous integer

**3541** is the smallest number whose reciprocal has period 20

**3542** is the number of ways to write 16 as an ordered sum of positive integers , where adjacent numbers are different

**3543** has a cube containing only 3 different digits

**3552** is a value of n for which n φ (n) is a palindrome

**3563** is a house number

**3564** divides 1^{1} + 2^{2} + 3^{3} + ^{ . . .} + 3564^{3564}

**3570** is both a triangular number and 6 times a triangular number

**3571** is the 17^{th} Lucas number

**3577** is a Kaprekar constant in base 2

**3579** has digits in arithmetic sequence

**3583** is the smallest number requiring an addition chain of length 16

**3584** is not the sum of 4 non-zero squares

**3585** has a 10^{th} power that contains the same digits as 9036^{9}

**3588** is the maximum number of regions space can be divided into by 23 spheres

**3593** is a prime that is the average of two 4^{th} powers

**3594** is the smallest number whose 9^{th} power has 32 digits

**3596** is the number permutations of {1,2,3,...,19} where adjacent numbers differ by no more than 2

**3599** is the product of twin primes

**3600** is the order of a perfect group (A060793)

**3605** is a centered tetrahedral number

**3607** is a prime factor of 123456789

**3609** is the number of multigraphs with 22 vertices and 4 edges

**3610** is a value of n for which n! - 1 is prime

**3612** is a narcissistic number in base 7

**3613** is a narcissistic number in base 7

**3616** = 1111 in base 15

**3620** is the trinomial coefficient T(16,12)

**3622** is the number of ways of placing 26 points on a 13×13 grid so that no 3 points are on a line

**3623** times the 3623^{th} prime is a palindrome .

**3624** is the first of five consecutive squareful numbers

**3626** is a member of the Fibonacci -type sequence starting with 1 and 9

**3630** appears inside its 4^{th} power

**3632** is a value of n for which n φ (n) is a palindrome

**3635** has a square with the first 3 digits the same as the next 3 digits

**3638** is the number of ways to stack 26 pennies in contiguous rows so that each penny lies on the table or on two pennies

**3640** = 13!!!

**3641** is an hexagonal prism number

**3645** is the maximum determinant of a binary 12×12 matrix

**3648** is the number of subsets of {1,2,3,...,15} that have a sum divisible by 9

**3650** is the number of binary cube-free words of length 19

**3654** = _{29}C _{3}

**3655** is the sum of consecutive squares in 2 ways

**3657** is a structured truncated octahedral number

**3658** is the number of forests with 13 vertices

**3663** is a palindrome in base 8 and in base 10

**3664** is the number of graphs with 10 vertices and 9 edges

**3665** would be prime if preceded and followed by a 1, 3, 7, or 9

**3671** is the number of 9-abolos

**3673** is the number of ways a 8×1 rectangle can be surrounded by 8×1 rectangles

**3678** has a square comprised of the digits 1-8

**3679** is the number of ways to stack 17 pennies in a line so that each penny lies on the table or on two pennies

**3681** is the maximum number of pieces a torus can be cut into with 27 cuts

**3683** is the maximum number of regions a cube can be cut into with 28 cuts

**3684** is a Keith number

**3685** is a strong Friedman number

**3686** would be prime if preceded and followed by a 1, 3, 7, or 9

**3691** is a number n for which n^{2}+1 is triple another square

**3696** is the number of ways to color the vertices of a square with 11 colors, up to rotation

**3697** is the smallest number in base 6 whose square contains the same digits in the same proportion

**3698** has a square comprised of the digits 0-7

**3699** is the rectilinear crossing number of complete graph K_{24}