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A379756 a(n) is the number of subsets of S(n) that sum to A023196(n), where S(n) is the set of the proper divisors (or aliquot parts) of A023196(n).

%I A379756 #7 Feb 15 2025 14:28:15
%S A379756 1,2,2,1,5,1,3,7,3,2,10,3,2,34,2,0,31,1,6,25,1,23,21,2,1,1,20,4,1,279,
%T A379756 13,15,1,15,116,9,11,12,4,197,1,2,755,1,42,2,9,12,6,2,151,169,7,1,9,8,
%U A379756 6,2190,1,516,1,6,121,130,1,6,119,1,469,4,446,1,4,6
%N A379756 a(n) is the number of subsets of S(n) that sum to A023196(n), where S(n) is the set of the proper divisors (or aliquot parts) of A023196(n).
%C A379756 This sequence is A065205 without the terms A065205(k) where k > sigma(k)/2.
%H A379756 Felix Huber, <a href="/A379756/b379756.txt">Table of n, a(n) for n = 1..10000</a>
%H A379756 Felix Huber, <a href="/A379756/a379756.txt">Maple program to calculate the distinct subsets</a>
%F A379756 Iff a(k) = 0, A023196(k) is a weird number (A006037).
%F A379756 Iff a(k) = 1, A023196(k) is a term of A064771.
%F A379756 a(A000396(k)) = 1 (A000396: perfect numbers).
%e A379756 a(8) = 7 because exactly the 7 subsets {6, 12, 18}, {3, 6, 9, 18}, {2, 4, 12, 18}, {2, 3, 4, 9, 18}, {2, 3, 4, 6, 9, 12}, {1, 2, 6, 9, 18}, {1, 2, 3, 12, 18} of S(8) = {1, 2, 3, 4, 6, 9, 12, 18} sum to A023196(8) = 36.
%e A379756 a(16) = 0 because no subset of S(16) = {1, 2, 5, 7, 10, 14, 35} sums to A023196(16) = 70 (weird number).
%p A379756 with(NumberTheory):
%p A379756 A023196:=proc(n)
%p A379756     local a;
%p A379756     option remember;
%p A379756     if n=1 then
%p A379756         6
%p A379756     else
%p A379756         for a from procname(n-1)+1 do
%p A379756             if sigma(a)>=2*a then
%p A379756                 return a
%p A379756             fi
%p A379756         od
%p A379756     fi;
%p A379756 end proc;
%p A379756 A379756:=proc(n)
%p A379756     local b,d,l;
%p A379756     d:=sigma(A023196(n))-2*A023196(n);
%p A379756     l:= [select(x->x<=d,Divisors(A023196(n)))[]];
%p A379756     b:= proc(m,i)
%p A379756         option remember;
%p A379756         `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
%p A379756     end proc;
%p A379756     forget(b);
%p A379756     b(d,nops(l))
%p A379756 end proc;
%p A379756 seq(A379756(n),n=1..74);
%Y A379756 Cf. A000203, A000396, A001065, A005101, A023196, A026793, A027750, A033630, A033882, A064771, A065205.
%K A379756 nonn
%O A379756 1,2
%A A379756 _Felix Huber_, Feb 07 2025