cp's OEIS Frontend

A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).

%I A320835 #12 Oct 23 2018 17:20:28
%S A320835 1,-1,0,0,-1,0,1,1,1,1,-1,1,1,0,0,1,-1,0,2,1,1,1,-2,0,1,0,0,0,2,0,-2,
%T A320835 -2,-1,1,-1,-2,3,-1,1,-2,-3,-2,3,0,-3,1,-4,-5,1,-1,-2,-1,5,-5,1,-3,1,
%U A320835 -1,-5,-4,5,1,-1,-9,-2,-1,-6,-1,-3,-2,7,-7,-8,-2,-2
%N A320835 a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).
%C A320835 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 A320835 Alois P. Heinz, <a href="/A320835/b320835.txt">Table of n, a(n) for n = 1..5000</a>
%F A320835 a(n) = A316441(A181821(n)).
%p A320835 with(numtheory):
%p A320835 b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
%p A320835       -add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
%p A320835     end:
%p A320835 a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
%p A320835          sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
%p A320835 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 23 2018
%t A320835 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A320835 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320835 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320835 Table[Sum[(-1)^(Length[m]-1),{m,mps[nrmptn[n]]}],{n,30}]
%Y A320835 Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320836.
%K A320835 sign,look
%O A320835 1,19
%A A320835 _Gus Wiseman_, Oct 21 2018
%E A320835 More terms from _Alois P. Heinz_, Oct 21 2018