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

0,5

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

Formula

a(n) = -Sum_{d|n, d<n} moebius(n/d)*A000041(d) = A000041(n) - A000837(n). - Vladeta Jovovic, Jun 17 2003

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 *)

Prog

(PARI) a(n) = - sumdiv(n, d, (d<n)*moebius(n/d)*numbpart(d)); \\ Michel Marcus, Oct 07 2017

Crossrefs

Cf. A000041, A000837, A083710.

Sequence in context: A214579 A083711 A339619 * A200976 A328187 A331885

Adjacent sequences: A018780 A018781 A018782 * A018784 A018785 A018786

Keyword

nonn

Author

David W. Wilson

Status

approved