A133732 A054525 * A000041.

1, 0, 1, 2, 4, 5, 10, 12, 20, 25, 41, 47, 76, 90, 129, 161, 230, 270, 384, 458, 615, 750, 1001, 1187, 1570, 1881, 2414, 2907, 3717, 4400, 5603, 6666, 8306, 9912, 12295, 14537, 17976, 21252, 25937, 30683, 37337, 43861, 53173, 62467, 75020, 88132

Offset: 1

Author

Gary W. Adamson, Sep 22 2007

Keywords

• nonn

Comments

A000041 = (1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...).

Examples

			a(4) = 2 = (0, -1, 0, 1) dot (1, 1, 2, 3) = (0, -1, 0, 3).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Mircea Merca and Maxie D. Schmidt, Generating Special Arithmetic Functions by Lambert Series Factorizations, arXiv:1706.00393 [math.NT], 2017. See Conjecture 3.1.
• Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.

Crossrefs

Cf. A054525.

Programs

Maple

read("transforms") : A000041 := proc(n) combinat[numbpart](n) ; end: a000041 := [seq(A000041(n),n=0..150)] ; a133732 := MOBIUS(a000041) ; # R. J. Mathar, Jan 19 2009
mob := (m,n) -> if irem(m,n) = 0 then numtheory:-mobius(m/n) else 0 fi:
A133732 := n -> add(mob(n,d)*combinat:-numbpart(d-1), d=1..n):
seq(A133732(n), n=1..46); # Peter Luschny, Jan 20 2018

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#]*PartitionsP[#-1]&];
Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Jan 20 2018 *)

Formula

Möbius transform of A000041, the partition numbers.

Extensions

More terms from R. J. Mathar, Jan 19 2009

Offset set to 1 by Peter Luschny, Jan 20 2018