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

Author

Gus Wiseman, Sep 19 2019

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

See link for additional cross-references.

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

Programs

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}]

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

Extensions

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