A182270 - OEIS

%I #31 Sep 23 2019 14:45:03

%S 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 73,32,99,53,125,60,186,92,215,127,311,164,394,221,518,320,656,386,

%U 903,545,1091,719,1470,925,1863,1215,2390,1642,3015,2037,3966

%N 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 Alois P. Heinz, <a href="/A182270/b182270.txt">Table of n, a(n) for n = 0..10000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Euler transform of A038548-1.

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

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

%e a(0) = 1: 0 = the empty sum.

%e a(1) = a(2) = a(3) = 0: no product is < 4.

%e a(4) = 1: 4 = 2*2.

%e a(6) = 1: 6 = 2*3.

%e a(8) = 2: 8 = 2*2 + 2*2 = 2*4.

%e a(9) = 1: 9 = 3*3.

%e a(12) = 5: 12 = 2*2 + 2*2 + 2*2 = 2*2 + 2*4 = 2*3 + 2*3 = 2*6 = 3*4.

%e a(13) = 1: 13 = 2*2 + 3*3.

%e a(14) = 4: 14 = 2*2 + 2*2 + 2*3 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p        d*(ceil(tau(d)/2)-1), d=divisors(j)) *a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..70);

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

%K nonn

%O 0,9

%A _Alois P. Heinz_, Apr 22 2012