252 (number) explained

Number:	252
Divisor:	1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252

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

In mathematics

252 is:

\tbinom{ 10 }{ 5 } , the largest one divisible by all coefficients in the previous line^[1] \tau( 3 ) , where \tau is the Ramanujan tau function.^[2] \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.}

• a practical number,^[4]
• a refactorable number,^[5]
• a hexagonal pyramidal number.^[6]
• a member of the Mian-Chowla sequence.^[7]

There are 252 points on the surface of a cuboctahedron of radius five in the face-centered cubic lattice,^[8] 252 ways of writing the number 4 as a sum of six squares of integers,^[9] 252 ways of choosing four squares from a 4×4 chessboard up to reflections and rotations,^[10] and 252 ways of placing three pieces on a Connect Four board.^[11]

Notes and References

1. Central binomial coefficients.
2. Ramanujan's tau function.
3. sigma_3(n): sum of cubes of divisors of n.
4. Practical numbers.
5. Web site: Sloane's A033950 : Refactorable numbers. 2016-04-18. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-04-18.
6. Hexagonal pyramidal numbers, or greengrocer's numbers.
7. Web site: Sloane's A005282 : Mian-Chowla sequence. 2016-04-19. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-04-19.
8. Number of points on surface of cuboctahedron.
9. Number of ways of writing n as a sum of 6 squares.
10. Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same.
11. Number of possible positions for n men on a standard 7 X 6 board of Connect-Four.