A320836 - OEIS

%I #11 Oct 23 2018 18:04:02

%S 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 -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 -4,-1,-1,0,-5,0,-1,-3,-9,-2,-3,0,-2,-1,-3,0,-7,0

%N 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 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 Alois P. Heinz, <a href="/A320836/b320836.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = A114592(A181821(n)).

%p with(numtheory):

%p b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,

%p       -add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))

%p     end:

%p a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(

%p          sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 23 2018

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t Table[Sum[(-1)^Length[m],{m,Select[mps[nrmptn[n]],UnsameQ@@#&]}],{n,30}]

%Y Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320835.

%K sign,look

%O 1,27

%A _Gus Wiseman_, Oct 21 2018

%E More terms from _Alois P. Heinz_, Oct 21 2018