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A061555 Integer part of sigma(n!)/n!.

%I A061555 #30 Apr 16 2024 22:03:04
%S A061555 1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,
%T A061555 6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A061555 7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8
%N A061555 Integer part of sigma(n!)/n!.
%C A061555 With increasing n, a(n) goes to infinity (proof in Sierpiński).
%C A061555 From _Bernard Schott_, Oct 03 2022: (Start)
%C A061555 It seems that sigma(n!)/n! is an integer only for n = 0, 1, 3, 5 and corresponding values are 1, 1, 2, 3.
%C A061555 For m >= 2, the smallest integer n such that a(n) = m is A061556(m). (End)
%D A061555 Wacław Sierpiński, Elementary Theory of Numbers, Ex. 6, p. 169, Warsaw, 1964.
%H A061555 Harry J. Smith, <a href="/A061555/b061555.txt">Table of n, a(n) for n = 0..1000</a>
%F A061555 a(n) = floor(sigma(n!)/n!) = floor(A062569(n)/A000142(n)).
%t A061555 Table[Floor[DivisorSigma[1, n!]/n!], {n, 0, 100}] (* _Wesley Ivan Hurt_, Apr 16 2024 *)
%o A061555 (PARI) { for (n=0, 1000, write("b061555.txt", n, " ", sigma(n!)\n!) ) } \\ _Harry J. Smith_, Jul 24 2009
%Y A061555 Cf. A000142, A000203, A061556, A062569.
%K A061555 nonn
%O A061555 0,4
%A A061555 _Labos Elemer_, May 17 2001
%E A061555 Terms corrected for an offset of 0 by _Harry J. Smith_, Jul 24 2009