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

1, 4, 7, 15, 19, 39, 46, 80, 106, 160, 201, 318, 390, 554, 729, 998, 1262, 1727, 2168, 2894, 3670, 4749, 5963, 7737, 9635, 12232, 15257, 19206, 23727, 29723, 36509, 45296, 55512, 68292, 83298, 102079, 123805, 150697, 182254, 220790, 265766

Offset: 1

Author

Vladeta Jovovic, Feb 16 2004

Keywords

• easy
• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..9000

Crossrefs

Cf. A092314, A092322, A092269, A092321, A092313, A092310, A092311, A092268.

Cf. A046746.

Programs

Maple

b:= proc(n, i) option remember; `if`(irem(n, i)=0, n, 0)
       +`if`(i>1, add(b(n-i*j, i-1), j=0..(n-1)/i), 0)
    end:
a:= n-> b(n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 04 2016

Mathematica

ss[n_]:=Module[{m=Min[n]},Select[n,#==m&]]; Table[Total[Flatten[ss/@ IntegerPartitions[n]]],{n,50}] (* Harvey P. Dale, Dec 16 2013 *)
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 *)

Formula

G.f.: Sum(n*x^n/(1-x^n)*Product(1/(1-x^k), k = n .. infinity), n = 1 .. infinity).

a(n) ~ sqrt(2) * exp(Pi*sqrt(2*n/3)) / (4*Pi*sqrt(n)). - Vaclav Kotesovec, Jul 06 2019

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004