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

0,3

Comments

See also A062569 sigma_1(n!).

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.

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A357333 A088992 A320502 * A051893 A354685 A247951

Adjacent sequences: A116903 A116904 A116905 * A116907 A116908 A116909

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 15 2006

Status

approved