A182270 Number of representations of n as a sum of products of pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 5, 1, 4, 2, 9, 2, 11, 3, 16, 7, 19, 6, 34, 13, 35, 18, 57, 23, 73, 32, 99, 53, 125, 60, 186, 92, 215, 127, 311, 164, 394, 221, 518, 320, 656, 386, 903, 545, 1091, 719, 1470, 925, 1863, 1215, 2390, 1642, 3015, 2037, 3966

Offset

0,9

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Transforms

Formula

Euler transform of A038548-1.

G.f.: Product_{k>0} 1/(1-x^k)^(A038548(k)-1).

G.f.: Product_{i>=1} Product_{j=2..i} 1/(1 - x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019

Example

a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 2: 8 = 2*2 + 2*2 = 2*4.
a(9) = 1: 9 = 3*3.
a(12) = 5: 12 = 2*2 + 2*2 + 2*2 = 2*2 + 2*4 = 2*3 + 2*3 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(14) = 4: 14 = 2*2 + 2*2 + 2*3 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
       d*(ceil(tau(d)/2)-1), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*(Ceiling[DivisorSigma[0, d]/2] - 1), {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)

Crossrefs

Cf. A000005, A006171, A038548, A066739, A182269, A211856, A211857.

Sequence in context: A308209 A366403 A212215 * A246272 A055135 A334873

Adjacent sequences: A182267 A182268 A182269 * A182271 A182272 A182273

Keyword

nonn

Author

Alois P. Heinz, Apr 22 2012

Status

approved