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A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

%I A047966 #36 Oct 27 2023 19:19:37
%S A047966 1,2,3,4,4,8,6,10,11,15,13,25,19,29,33,42,39,62,55,81,84,103,105,153,
%T A047966 146,185,203,253,257,344,341,432,463,552,594,747,761,920,1003,1200,
%U A047966 1261,1537,1611,1921,2089,2410,2591,3095,3270,3815,4138,4769,5121,5972,6394,7367,7974,9066,9793,11305,12077,13736,14940
%N A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.
%C A047966 Number of partitions of n such that every part occurs with the same multiplicity. - _Vladeta Jovovic_, Oct 22 2004
%C A047966 Christopher and Christober call such partitions uniform. - _Gus Wiseman_, Apr 16 2018
%C A047966 Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  _Gary W. Adamson_, Jun 08 2009
%H A047966 Alois P. Heinz, <a href="/A047966/b047966.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A047966 A. David Christopher and M. Davamani Christober, <a href="http://emis.impa.br/EMIS/journals/GMN/yahoo_site_admin/assets/docs/1_GMN-2492-V13N2.77213831.pdf">Relatively Prime Uniform Partitions</a>, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.
%F A047966 G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - _Vladeta Jovovic_, Jun 22 2003
%F A047966 G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - _Ilya Gutkovskiy_, Jun 20 2018
%F A047966 a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2018
%e A047966 The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018
%p A047966 with(numtheory):
%p A047966 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A047966      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
%p A047966     end:
%p A047966 a:= n-> add(b(d), d=divisors(n)):
%p A047966 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 11 2016
%t A047966 b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Dec 06 2016 after _Alois P. Heinz_ *)
%t A047966 Table[DivisorSum[n,PartitionsQ],{n,20}] (* _Gus Wiseman_, Apr 16 2018 *)
%o A047966 (PARI)
%o A047966 N = 66; q='q+O('q^N);
%o A047966 D(q)=eta(q^2)/eta(q); \\ A000009
%o A047966 Vec( sum(e=1,N,D(q^e)-1) ) \\ _Joerg Arndt_, Mar 27 2014
%Y A047966 Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.
%K A047966 nonn
%O A047966 1,2
%A A047966 _N. J. A. Sloane_