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A293813 Number of partitions of n into nontrivial divisors of n.

%I A293813 #11 Jun 14 2018 02:33:04
%S A293813 1,0,0,0,1,0,2,0,3,1,2,0,11,0,2,2,9,0,14,0,15,2,2,0,79,1,2,4,19,0,93,
%T A293813 0,35,2,2,2,279,0,2,2,157,0,153,0,27,24,2,0,1075,1,28,2,31,0,254,2,
%U A293813 261,2,2,0,7025,0,2,31,201,2,320,0,39,2,301,0,12071,0,2,35,43,2,427,0,3073
%N A293813 Number of partitions of n into nontrivial divisors of n.
%H A293813 Antti Karttunen, <a href="/A293813/b293813.txt">Table of n, a(n) for n = 0..10000</a> (computed from the b-file of A211110 provided by Alois P. Heinz)
%H A293813 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A293813 a(n) = [x^n] Product_{d|n, 1 < d < n} 1/(1 - x^d).
%F A293813 a(n) = A211110(n) - 1 for n > 1.
%e A293813 a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].
%p A293813 with(numtheory):
%p A293813 a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
%p A293813       b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A293813              b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A293813           end; forget(b):
%p A293813       b(n, nops(l))
%p A293813     end:
%p A293813 seq(a(n), n=0..100);  # _Alois P. Heinz_, Oct 16 2017
%t A293813 Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 80}]
%Y A293813 Cf. A018818, A027750, A070824, A210442, A211110, A293814.
%K A293813 nonn
%O A293813 0,7
%A A293813 _Ilya Gutkovskiy_, Oct 16 2017