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A092309 Sum of smallest parts (counted with multiplicity) of all partitions of n.

%I A092309 #19 Jul 06 2019 04:42:57
%S A092309 1,4,7,15,19,39,46,80,106,160,201,318,390,554,729,998,1262,1727,2168,
%T A092309 2894,3670,4749,5963,7737,9635,12232,15257,19206,23727,29723,36509,
%U A092309 45296,55512,68292,83298,102079,123805,150697,182254,220790,265766
%N A092309 Sum of smallest parts (counted with multiplicity) of all partitions of n.
%H A092309 Alois P. Heinz, <a href="/A092309/b092309.txt">Table of n, a(n) for n = 1..9000</a>
%F A092309 G.f.: Sum(n*x^n/(1-x^n)*Product(1/(1-x^k), k = n .. infinity), n = 1 .. infinity).
%F A092309 a(n) ~ sqrt(2) * exp(Pi*sqrt(2*n/3)) / (4*Pi*sqrt(n)). - _Vaclav Kotesovec_, Jul 06 2019
%e A092309 Partitions of 4 are: [1,1,1,1], [1,1,2], [2,2], [1,3], [4]; thus a(4)=4*1+2*1+2*2+1*1+1*4=15.
%p A092309 b:= proc(n, i) option remember; `if`(irem(n, i)=0, n, 0)
%p A092309        +`if`(i>1, add(b(n-i*j, i-1), j=0..(n-1)/i), 0)
%p A092309     end:
%p A092309 a:= n-> b(n$2):
%p A092309 seq(a(n), n=1..50);  # _Alois P. Heinz_, Feb 04 2016
%t A092309 ss[n_]:=Module[{m=Min[n]},Select[n,#==m&]]; Table[Total[Flatten[ss/@ IntegerPartitions[n]]],{n,50}] (* _Harvey P. Dale_, Dec 16 2013 *)
%t A092309 b[n_, i_] := b[n, i] = If[Mod[n, i] == 0, n, 0] + If[i > 1, Sum[b[n - i*j, i - 1], {j, 0, (n - 1)/i}], 0]; a[n_] := b[n, n]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y A092309 Cf. A092314, A092322, A092269, A092321, A092313, A092310, A092311, A092268.
%Y A092309 Cf. A046746.
%K A092309 easy,nonn
%O A092309 1,2
%A A092309 _Vladeta Jovovic_, Feb 16 2004
%E A092309 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004