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A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).

%I A320836 #11 Oct 23 2018 18:04:02
%S A320836 1,-1,-1,0,0,1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,-1,-1,0,-2,-1,0,-2,0,
%T A320836 -2,-1,-1,-1,-4,-1,-1,-1,-3,0,-3,0,-2,-4,-1,-1,-6,-2,-3,-2,-2,0,-6,-2,
%U A320836 -4,-1,-1,0,-5,0,-1,-3,-9,-2,-3,0,-2,-1,-3,0,-7,0
%N A320836 a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).
%C A320836 This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%H A320836 Alois P. Heinz, <a href="/A320836/b320836.txt">Table of n, a(n) for n = 1..5000</a>
%F A320836 a(n) = A114592(A181821(n)).
%p A320836 with(numtheory):
%p A320836 b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
%p A320836       -add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
%p A320836     end:
%p A320836 a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
%p A320836          sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
%p A320836 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 23 2018
%t A320836 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A320836 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320836 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320836 Table[Sum[(-1)^Length[m],{m,Select[mps[nrmptn[n]],UnsameQ@@#&]}],{n,30}]
%Y A320836 Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320835.
%K A320836 sign,look
%O A320836 1,27
%A A320836 _Gus Wiseman_, Oct 21 2018
%E A320836 More terms from _Alois P. Heinz_, Oct 21 2018