A327657 Number of divisors of n that are 1 or whose prime indices have a common divisor > 1.

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 3, 2, 1, 3, 2, 3, 3, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 3, 3, 2, 2, 4, 3, 2, 4, 2, 2, 3, 2, 2, 6, 1, 4, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 2, 3, 4, 2, 2, 5, 2, 2, 4, 3, 2, 4, 2, 2, 4, 4, 2, 3, 2, 3, 2, 2, 3, 4, 3, 2, 3, 2, 2, 5

Offset: 1

Author

Gus Wiseman, Sep 21 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 whose prime indices have a common divisor > 1 are listed in A318978.

Examples

			The divisors of 90 that are 1 or whose prime indices have a common divisor > 1 are {1, 3, 5, 9}, so a(90) = 4.
		

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, A281116, A289509, A318979, A327406, A327656.

Programs

Mathematica

Table[Length[Select[Divisors[n],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]],{n,100}]

PARI

A327657(n) = sumdiv(n, d, (1==d)||(gcd(apply(x->primepi(x), factor(d)[, 1]))>1)); \\ Antti Karttunen, Dec 05 2021

Formula

a(n) = A000005(n) - A318979(n). - Antti Karttunen, Dec 05 2021

Extensions

Data section extended up to 105 terms by Antti Karttunen, Dec 05 2021