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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.

%I A182270 #31 Sep 23 2019 14:45:03
%S A182270 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,
%T A182270 73,32,99,53,125,60,186,92,215,127,311,164,394,221,518,320,656,386,
%U A182270 903,545,1091,719,1470,925,1863,1215,2390,1642,3015,2037,3966
%N 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.
%H A182270 Alois P. Heinz, <a href="/A182270/b182270.txt">Table of n, a(n) for n = 0..10000</a>
%H A182270 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A182270 Euler transform of A038548-1.
%F A182270 G.f.: Product_{k>0} 1/(1-x^k)^(A038548(k)-1).
%F A182270 G.f.: Product_{i>=1} Product_{j=2..i} 1/(1 - x^(i*j)). - _Ilya Gutkovskiy_, Sep 23 2019
%e A182270 a(0) = 1: 0 = the empty sum.
%e A182270 a(1) = a(2) = a(3) = 0: no product is < 4.
%e A182270 a(4) = 1: 4 = 2*2.
%e A182270 a(6) = 1: 6 = 2*3.
%e A182270 a(8) = 2: 8 = 2*2 + 2*2 = 2*4.
%e A182270 a(9) = 1: 9 = 3*3.
%e A182270 a(12) = 5: 12 = 2*2 + 2*2 + 2*2 = 2*2 + 2*4 = 2*3 + 2*3 = 2*6 = 3*4.
%e A182270 a(13) = 1: 13 = 2*2 + 3*3.
%e A182270 a(14) = 4: 14 = 2*2 + 2*2 + 2*3 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.
%p A182270 with(numtheory):
%p A182270 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A182270        d*(ceil(tau(d)/2)-1), d=divisors(j)) *a(n-j), j=1..n)/n)
%p A182270     end:
%p A182270 seq(a(n), n=0..70);
%t A182270 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_ *)
%Y A182270 Cf. A000005, A006171, A038548, A066739, A182269, A211856, A211857.
%K A182270 nonn
%O A182270 0,9
%A A182270 _Alois P. Heinz_, Apr 22 2012