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A116906 Sum of squares of divisors of n!.

%I A116906 #21 Dec 10 2024 17:59:08
%S A116906 1,1,5,50,850,22100,806806,40340300,2584263500,209609328500,
%T A116906 20993420690550,2561197324247100,368819285671473000,
%U A116906 62699278564150410000,12294076739210974071000,2766341857504878226501200
%N A116906 Sum of squares of divisors of n!.
%C A116906 See also A062569 sigma_1(n!).
%H A116906 Amiram Eldar, <a href="/A116906/b116906.txt">Table of n, a(n) for n = 0..253</a>
%H A116906 Jean-Marie De Koninck and William Verreault, <a href="https://doi.org/10.2298/PIM2429045D">Arithmetic functions at factorial arguments</a>, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.
%F A116906 a(n) = A001157(A000142(n)).
%F A116906 a(n) = Sum_{d|n!} d^2.
%F A116906 a(n) = sigma_2(n!).
%F A116906 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
%e A116906 a(0) = 1 because only 1 divides 0! = 1.
%e A116906 a(1) = 1 because only 1 divides 1! = 1.
%e A116906 a(2) = 5 because both 1 and 2 divide 2! = 2 and 1^2 + 2^2 = 5.
%e A116906 a(3) = 50 because 1, 2, 3, 6 divide 3! = 6 and 1^2 + 2^2 + 3^2 + 6^2 = 50.
%e A116906 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.
%t A116906 a[n_] := DivisorSigma[2, n!]; Array[a, 16, 0] (* _Amiram Eldar_, Aug 01 2019 *)
%o A116906 (Sage) [sigma(factorial(n),2)for n in range(0,16)] # _Zerinvary Lajos_, Jun 13 2009
%Y A116906 Cf. A000142, A000203, A001157, A013661, A062569.
%K A116906 easy,nonn
%O A116906 0,3
%A A116906 _Jonathan Vos Post_, Mar 15 2006