A018783 Number of partitions of n into parts having a common factor.

0, 0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 1, 38, 1, 45, 17, 57, 1, 94, 7, 102, 30, 138, 1, 218, 1, 231, 58, 298, 21, 451, 1, 491, 103, 644, 1, 919, 1, 1005, 203, 1256, 1, 1784, 15, 1993, 299, 2439, 1, 3365, 62, 3735, 492, 4566, 1, 6252, 1, 6843, 819, 8349, 107, 11096

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8

Crossrefs

Cf. A000041, A000837, A083710.

Programs

Maple

with(numtheory): with(combinat):
a:= n-> `if`(n=0, 0,
         numbpart(n) -add(mobius(n/d)*numbpart(d), d=divisors(n))):
seq(a(n), n=0..100); # Alois P. Heinz, Nov 29 2011

Mathematica

A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; a[0] = 0; a[n_] := PartitionsP[n] - A000837[n]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Oct 03 2013, after Vladeta Jovovic *)

PARI

a(n) = - sumdiv(n, d, (dMichel Marcus, Oct 07 2017

Formula

a(n) = -Sum_{d|n, dA000041(d) = A000041(n) - A000837(n). - Vladeta Jovovic, Jun 17 2003