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

1,2

Links

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

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

Example

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.

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 *)

Crossrefs

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

Cf. A046746.

Sequence in context: A310926 A310927 A049832 * A263617 A271675 A356714

Adjacent sequences: A092306 A092307 A092308 * A092310 A092311 A092312

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 16 2004

Extensions

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

Status

approved