A327513 Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.

1, 2, 1, 3, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 2, 5, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 6, 1, 6, 2, 3, 2, 5, 1, 3, 1, 7, 1, 4, 1, 5, 2, 3, 1, 9, 1, 3, 2, 5, 1, 3, 2, 7, 1, 3, 1, 10, 1, 3, 1, 7, 1, 6, 1, 5, 2, 6, 1, 7, 1, 3, 2, 5, 2, 4, 1, 9, 1, 3, 1, 7, 2, 3, 1, 7, 1, 6, 1, 5, 2, 3, 2, 11, 1, 3, 2, 5, 1, 6, 1, 7, 3

Offset

1,2

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. Numbers that are 1, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696.

Note that the maximum odd divisor of any entry must be squarefree.

Number of terms of A302696 that divide n. Put in other words, this sequence is the inverse Möbius transform of the characteristic function of A302696. - Antti Karttunen, Dec 06 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Index entries for sequences computed from indices in prime factorization

Example

The divisors of 72 that are 1, 2, or nonprime numbers whose prime indices are pairwise coprime are: {1, 2, 4, 6, 8, 12, 24}, so a(72) = 7.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Divisors[n], #==1||CoprimeQ@@primeMS[#]&]], {n, 100}]

Prog

(PARI)
isA302696(n) = if(isprimepower(n), !(n%2), if(!issquarefree(n>>valuation(n, 2)), 0, my(pis=apply(primepi, factor(n)[, 1])); (lcm(pis)==factorback(pis))));
A327513(n) = sumdiv(n, d, isA302696(d)); \\ Antti Karttunen, Dec 06 2021

Crossrefs

See link for additional cross-references.

Cf. A000005, A056239, A112798, A302569, A302696, A304711.

Sequence in context: A326154 A306248 A361788 * A327530 A084360 A082460

Adjacent sequences: A327510 A327511 A327512 * A327514 A327515 A327516

Keyword

nonn

Author

Gus Wiseman, Sep 19 2019

Extensions

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021

Status

approved