A116906 Sum of squares of divisors of n!.

1, 1, 5, 50, 850, 22100, 806806, 40340300, 2584263500, 209609328500, 20993420690550, 2561197324247100, 368819285671473000, 62699278564150410000, 12294076739210974071000, 2766341857504878226501200

Offset: 0

Author

Jonathan Vos Post, Mar 15 2006

Keywords

• easy
• nonn

Comments

See also A062569 sigma_1(n!).

Examples

			a(0) = 1 because only 1 divides 0! = 1.
a(1) = 1 because only 1 divides 1! = 1.
a(2) = 5 because both 1 and 2 divide 2! = 2 and 1^2 + 2^2 = 5.
a(3) = 50 because 1, 2, 3, 6 divide 3! = 6 and 1^2 + 2^2 + 3^2 + 6^2 = 50.
a(4) = 850 because 1, 2, 3, 4, 6, 8, 12, 24 divide 4! = 24 and 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 8^2 + 12^2 + 24^2 = 850.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..253
• Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Crossrefs

Cf. A000142, A000203, A001157, A013661, A062569.

Programs

Mathematica

a[n_] := DivisorSigma[2, n!]; Array[a, 16, 0] (* Amiram Eldar, Aug 01 2019 *)

Sage

[sigma(factorial(n),2)for n in range(0,16)] # Zerinvary Lajos, Jun 13 2009

Formula

a(n) = A001157(A000142(n)).

a(n) = Sum_{d|n!} d^2.

a(n) = sigma_2(n!).

a(n) = zeta(2) * n!^2 * (1 + O(log(n)/n)) (De Koninck and Verreault, 2024. p. 54, Theorem 4.5). - Amiram Eldar, Dec 10 2024