A062569 a(n) = sigma(n!).
1, 1, 3, 12, 60, 360, 2418, 19344, 159120, 1481040, 15334088, 184009056, 2217441408, 31044179712, 442487616480, 6686252969760, 107004539285280, 1926081707135040, 34683832925921088, 693676658518421760, 13891399238731734720, 292460416142501376000
Offset: 0
Examples
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
References
- Wacław Sierpiński, Elementary Theory of Numbers, Ex. 6, p. 169, Warsaw, 1964.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
- Rafael Jakimczuk, Two topics in number theory: sum of divisors of the factorial and a formula for primes, International Mathematical Forum, Vol. 12, No. 19 (2017), pp. 929-935. See Theorem 1.4, p. 932.
- Vaclav Kotesovec, Plot of a(n)/(n!*log(n)) for n = 2..50000.
Programs
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Maple
with(numtheory):seq(sigma(n!), n=0..19); # Zerinvary Lajos, Feb 15 2008
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Mathematica
Array[DivisorSigma[1,#! ]&,33,1] (* Vladimir Joseph Stephan Orlovsky, Nov 01 2009 *)
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PARI
for(n=0,21,print(sigma(n!)))
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Sage
[sigma(ZZ(n).factorial(), 1) for n in range(20)] # Zerinvary Lajos, Jun 13 2009
Formula
a(p) = (p+1)*a(p-1) for p prime. - Altug Alkan, Jul 18 2016
Limit_{n->oo} a(n)/n! = oo. Proof in Sierpiński. - Bernard Schott, Feb 09 2019
a(n) ~ c * n! * log(n) * (1 + O(1/log(n))), where c = exp(gamma) = A073004 (Jakimczuk, 2017). - Amiram Eldar, Nov 07 2020