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

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, 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, 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

Offset: 0

Author

Labos Elemer, May 17 2001

Keywords

• nonn

Comments

With increasing n, a(n) goes to infinity (proof in Sierpiński).

From Bernard Schott, Oct 03 2022: (Start)

It seems that sigma(n!)/n! is an integer only for n = 0, 1, 3, 5 and corresponding values are 1, 1, 2, 3.

For m >= 2, the smallest integer n such that a(n) = m is A061556(m). (End)

References

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

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000142, A000203, A061556, A062569.

Programs

Mathematica

Table[Floor[DivisorSigma[1, n!]/n!], {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

PARI

{ for (n=0, 1000, write("b061555.txt", n, " ", sigma(n!)\n!) ) } \\ Harry J. Smith, Jul 24 2009

Formula

a(n) = floor(sigma(n!)/n!) = floor(A062569(n)/A000142(n)).

Extensions

Terms corrected for an offset of 0 by Harry J. Smith, Jul 24 2009