A133732 - OEIS

%I #27 Jul 21 2020 22:14:02

%S 1,0,1,2,4,5,10,12,20,25,41,47,76,90,129,161,230,270,384,458,615,750,

%T 1001,1187,1570,1881,2414,2907,3717,4400,5603,6666,8306,9912,12295,

%U 14537,17976,21252,25937,30683,37337,43861,53173,62467,75020,88132

%N A054525 * A000041.

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

%H Michael De Vlieger, <a href="/A133732/b133732.txt">Table of n, a(n) for n = 1..10000</a>

%H Mircea Merca and Maxie D. Schmidt, <a href="https://arxiv.org/abs/1706.00393">Generating Special Arithmetic Functions by Lambert Series Factorizations</a>, arXiv:1706.00393 [math.NT], 2017. See Conjecture 3.1.

%H Maxie Dion Schmidt, <a href="https://arxiv.org/abs/2004.02976">A catalog of interesting and useful Lambert series identities</a>, arXiv:2004.02976 [math.NT], 2020.

%F Möbius transform of A000041, the partition numbers.

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

%p 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

%p mob := (m,n) -> if irem(m,n) = 0 then numtheory:-mobius(m/n) else 0 fi:

%p A133732 := n -> add(mob(n,d)*combinat:-numbpart(d-1), d=1..n):

%p seq(A133732(n), n=1..46); # _Peter Luschny_, Jan 20 2018

%t a[n_] := DivisorSum[n, MoebiusMu[n/#]*PartitionsP[#-1]&];

%t Table[a[n], {n, 1, 45}] (* _Jean-François Alcover_, Jan 20 2018 *)

%Y Cf. A054525.

%K nonn

%O 1,4

%A _Gary W. Adamson_, Sep 22 2007

%E More terms from _R. J. Mathar_, Jan 19 2009

%E Offset set to 1 by _Peter Luschny_, Jan 20 2018