A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

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

Offset

1,6

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Index entries for sequences related to prime indices in the factorization of n.

Example

The a(60) = 8 factorizations: (2*5*6), (3*4*5), (2*30), (3*20), (4*15), (5*12), (6*10), (60).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], UnsameQ@@Mean/@primeMS/@#&]], {n, 100}]

Prog

(PARI)
avgpis(n) = { my(f=factor(n)); f[, 1] = apply(primepi, f[, 1]); (1/bigomega(n))*sum(i=1, #f~, f[i, 2]*f[i, 1]); };
all_have_different_average_of_pis(facs) = if(!#facs, 1, (#Set(apply(avgpis, facs)) == #facs));
A326516(n, m=n, facs=List([])) = if(1==n, all_have_different_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A326516(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Crossrefs

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326513, A326514, A326515, A326521, A326537.

Sequence in context: A379141 A351414 A349056 * A081707 A368414 A303707

Adjacent sequences: A326513 A326514 A326515 * A326517 A326518 A326519

Keyword

nonn

Author

Gus Wiseman, Jul 12 2019

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

Status

approved