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).

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, -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, -1, -5, -4, 5, 1, -1, -9, -2, -1, -6, -1, -3, -2, 7, -7, -8, -2, -2

Offset

1,19

Comments

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}.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Formula

a(n) = A316441(A181821(n)).

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, -1)+`if`(isprime(n), 0,
      -add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, b(((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
         sort(map(i-> pi(i[1])$i[2], ifactors(n)[2]), `>`)))$2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2018

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[(-1)^(Length[m]-1), {m, mps[nrmptn[n]]}], {n, 30}]

Crossrefs

Cf. A001055, A001222, A007716, A045778, A114592, A162247, A181821, A305936, A316441, A318284, A319237, A319238, A320836.

Sequence in context: A364891 A143277 A292378 * A276183 A056563 A088231

Adjacent sequences: A320832 A320833 A320834 * A320836 A320837 A320838

Keyword

sign,look

Author

Gus Wiseman, Oct 21 2018

Extensions

More terms from Alois P. Heinz, Oct 21 2018

Status

approved