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a(n) = sigma(n!).
26

%I #60 Jan 23 2023 09:11:24

%S 1,1,3,12,60,360,2418,19344,159120,1481040,15334088,184009056,

%T 2217441408,31044179712,442487616480,6686252969760,107004539285280,

%U 1926081707135040,34683832925921088,693676658518421760,13891399238731734720,292460416142501376000

%N a(n) = sigma(n!).

%D Wacław Sierpiński, Elementary Theory of Numbers, Ex. 6, p. 169, Warsaw, 1964.

%H Paul D. Hanna, <a href="/A062569/b062569.txt">Table of n, a(n) for n = 0..300</a>

%H Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.71088">Two topics in number theory: sum of divisors of the factorial and a formula for primes</a>, International Mathematical Forum, Vol. 12, No. 19 (2017), pp. 929-935. See Theorem 1.4, p. 932.

%H Vaclav Kotesovec, <a href="/A062569/a062569.jpg">Plot of a(n)/(n!*log(n)) for n = 2..50000</a>.

%F a(n) = A000203(A000142(n)). - _Michel Marcus_, Jan 10 2015

%F a(p) = (p+1)*a(p-1) for p prime. - _Altug Alkan_, Jul 18 2016

%F Limit_{n->oo} a(n)/n! = oo. Proof in Sierpiński. - _Bernard Schott_, Feb 09 2019

%F a(n) ~ c * n! * log(n) * (1 + O(1/log(n))), where c = exp(gamma) = A073004 (Jakimczuk, 2017). - _Amiram Eldar_, Nov 07 2020

%e a(4) = 60, since the sum of the positive divisors of 4! is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60. - _Timothy L. Tiffin_, Jan 22 2023

%p with(numtheory):seq(sigma(n!), n=0..19); # _Zerinvary Lajos_, Feb 15 2008

%t Array[DivisorSigma[1,#! ]&,33,1] (* _Vladimir Joseph Stephan Orlovsky_, Nov 01 2009 *)

%o (PARI) for(n=0,21,print(sigma(n!)))

%o (Sage) [sigma(ZZ(n).factorial(), 1) for n in range(20)] # _Zerinvary Lajos_, Jun 13 2009

%Y Cf. A000142, A000203, A027423, A073004.

%K easy,nonn

%O 0,3

%A _Jason Earls_, Jul 03 2001