A327656 Maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 5, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 1, 11, 17, 7, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 57, 29, 59, 5, 61, 31, 63, 1, 65, 11, 67, 17, 23, 7, 71, 9, 73

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, which is the union of this sequence without 1.

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) = 9.
		

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

The union consists of {1} followed by A318978.

See link for additional cross-references.

Cf. A000005, A006530, A056239, A112798, A281116, A289509, A327406, A327657.

Programs

Mathematica

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

PARI

A327656(n) = vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))); \\ Antti Karttunen, Dec 06 2021

Formula

a(n) = n/A327405(n).

n belongs to A318978 iff a(n) = 1.