A145519 a(n) = Sum_{k=1..n} A145518(n,k).

1, 2, 7, 19, 54, 134, 354, 838, 2057, 4794, 11232, 25412, 58075, 128670, 286152, 625829, 1365653, 2941088, 6331146, 13474533, 28642325, 60404681, 127082128, 265712673, 554608226, 1151374963, 2385950536, 4924685252, 10145267212, 20831428273, 42708248451

Offset: 0

Author

Tilman Neumann, Oct 12 2008

Keywords

• nonn

Comments

Row sums of A145518.

Also row sums of A129129, A215366.

a(n) = sum of the Heinz numbers of the partitions of n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the 3 partitions of 3, namely [3], [1,2], and [1,1,1] we get 5, 2*3=6, and 2*2*2=8, respectively; their sum is a(3) = 19. - Emeric Deutsch, Jun 09 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• More terms in A145518 and A145519

Crossrefs

Cf. A129129, A145518, A215366, A334434.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
      add(b(n-i*j, i-1)*ithprime(i)^j, j=0..iquo(n, i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 19 2013

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, Sum[b[n-i*j, i-1]*Prime[i]^j, {j, 0, Quotient[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Formula

G.f.: 1/Product_{i>=1}(1-prime(i)*x^i). - Vladeta Jovovic, Nov 09 2008

a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1 - prime(k)/2^k) = 50.412394245500690832088704444961002125578414895935257436317... . - Vaclav Kotesovec, Sep 10 2014, updated Apr 11 2020

Extensions

a(0) inserted by Alois P. Heinz, Feb 19 2013