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252 (two hundred [and] fifty-two) is the natural number following 251 and preceding 253.

252 is the central binomial coefficient \( \tbinom{10}{5} \),[1] and is \( \tau(3) \), where \( \tau \) is the Ramanujan tau function.[2] 252 is also \( \sigma_3(6) \), where \( \sigma_3 \) is the function that sums the cubes of the divisors of its argument:[3]

\( 1^3+2^3+3^3+6^3=(1^3+2^3)(1^3+3^3)=252. \)

It is a practical number,[4] and a hexagonal pyramidal number.[5] There are 252 points on the surface of a cuboctahedron of radius five in the fcc lattice,[6] 252 ways of writing the number 4 as a sum of six squares of integers,[7] 252 ways of choosing four squares from a 4×4 chessboard up to reflections and rotations,[8] and 252 ways of placing four pieces on a Connect Four board.[9]

References

"Sloane's A000984 : Central binomial coefficients", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A000594 : Ramanujan's tau function", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A001158 : sigma_3(n): sum of cubes of divisors of n", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A005153 : Practical numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A002412 : Hexagonal pyramidal numbers, or greengrocer's numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A005901 : Number of points on surface of cuboctahedron", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A000141 : Number of ways of writing n as a sum of 6 squares", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A019318 : Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
"Sloane's A090224 : Number of possible positions for n men on a standard 7 X 6 board of Connect-Four", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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