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A211111 Number of partitions of n into distinct divisors > 1 of n.

%I A211111 #23 Nov 18 2021 12:13:36
%S A211111 1,0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,1,3,1,1,1,
%T A211111 1,1,6,1,1,1,2,1,2,1,1,1,1,1,6,1,1,1,1,1,3,1,2,1,1,1,19,1,1,1,1,1,3,1,
%U A211111 1,1,1,1,16,1,1,1,1,1,2,1,4,1,1,1,14,1
%N A211111 Number of partitions of n into distinct divisors > 1 of n.
%C A211111 a(A136446(n)) > 1.
%H A211111 Alois P. Heinz, <a href="/A211111/b211111.txt">Table of n, a(n) for n = 0..10000</a> (terms n=1..1000 from Reinhard Zumkeller)
%e A211111 n=12: the divisors > 1 of 12 are {2,3,4,6,12}, there are exactly two subsets which sum up to 12, namely {12} and {2,4,6}, therefore a(12) = 2;
%e A211111 a(13) = #{13} = 1, because 13 is prime, having no other divisor > 1;
%e A211111 n=14: the divisors > 1 of 14 are {2,7,14}, {14} is the only subset summing up to 14, therefore a(14) = 1.
%p A211111 with(numtheory):
%p A211111 a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
%p A211111       b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A211111              b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i-1))))
%p A211111           end; forget(b):
%p A211111       b(n, nops(l))
%p A211111     end:
%p A211111 seq(a(n), n=0..100);  # _Alois P. Heinz_, Nov 18 2021
%t A211111 a[n_] := Count[IntegerPartitions[n, All, Divisors[n] // Rest], P_ /; Reverse[P] == Union[P]];
%t A211111 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 18 2021 *)
%o A211111 (Haskell)
%o A211111 a211111 n = p (tail $ a027750_row n) n where
%o A211111    p _  0 = 1
%o A211111    p [] _ = 0
%o A211111    p (k:ks) m | m < k     = 0
%o A211111                | otherwise = p ks (m - k) + p ks m
%Y A211111 Cf. A211110, A033630, A027750.
%Y A211111 Cf. A065205, A136446.
%K A211111 nonn
%O A211111 0,13
%A A211111 _Reinhard Zumkeller_, Apr 01 2012
%E A211111 a(0)=1 prepended by _Alois P. Heinz_, Nov 18 2021