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A331979 Number of compositions (ordered partitions) of n into distinct nontrivial divisors of n.

%I A331979 #12 Nov 17 2020 05:40:52
%S A331979 1,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,30,0,0,0,0,0,30,0,0,
%T A331979 0,0,0,894,0,0,0,24,0,6,0,0,0,0,0,894,0,0,0,0,0,30,0,120,0,0,0,19518,
%U A331979 0,0,0,0,0,126,0,0,0,0,0,18558,0,0,0,0,0,6,0,864
%N A331979 Number of compositions (ordered partitions) of n into distinct nontrivial divisors of n.
%H A331979 Alois P. Heinz, <a href="/A331979/b331979.txt">Table of n, a(n) for n = 0..10000</a>
%H A331979 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%e A331979 a(12) = 6 because we have [6, 4, 2], [6, 2, 4], [4, 6, 2], [4, 2, 6], [2, 6, 4] and [2, 4, 6].
%p A331979 with(numtheory):
%p A331979 a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]):
%p A331979       b:= proc(m, i, p) option remember; `if`(m=0, p!, `if`(i<1, 0,
%p A331979              b(m, i-1, p)+`if`(l[i]>m, 0, b(m-l[i], i-1, p+1))))
%p A331979           end; forget(b):
%p A331979       b(n, nops(l), 0)
%p A331979     end:
%p A331979 seq(a(n), n=0..100);  # _Alois P. Heinz_, Feb 03 2020
%t A331979 a[n_] := If[n == 0, 1, Module[{b, l = Divisors[n] ~Complement~ {1, n}}, b[m_, i_, p_] := b[m, i, p] = If[m == 0, p!, If[i < 1, 0, b[m, i-1, p] + If[l[[i]] > m, 0, b[m - l[[i]], i-1, p+1]]]]; b[n, Length[l], 0]]];
%t A331979 a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%Y A331979 Cf. A018818, A027750, A033630, A065205, A070824, A100346, A211111, A293813, A293814, A294137, A294138, A331927, A331928.
%K A331979 nonn
%O A331979 0,13
%A A331979 _Ilya Gutkovskiy_, Feb 03 2020