A327657 -id:A327657 - cp's OEIS frontend

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

Gus Wiseman, Sep 21 2019

nonn

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.

			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.

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

The union consists of {1} followed by A318978.
See link for additional cross-references.
Cf. A000005, A006530, A056239, A112798, A281116, A289509, A327406, A327657.

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

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

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

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

A318979 Number of divisors of n with relatively prime prime indices, meaning they belong to A289509.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 1, 4, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 5, 0, 5, 1, 2, 1, 6, 0, 2, 0, 6, 0, 4, 0, 4, 2, 2, 0, 8, 0, 3, 1, 4, 0, 4, 1, 6, 0, 2, 0, 9, 0, 2, 0, 6, 0, 5, 0, 4, 1, 5, 0, 9, 0, 2, 2, 4, 1, 4, 0, 8, 0, 2, 0, 8, 1, 2, 0

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, corresponding to the prime index multisets (), (1), (2), (11), (12), (22), (112), (122), (1122) respectively. Of these, only (1), (11), (12), (112), (122), (1122) are relatively prime, corresponding to the divisors 2, 4, 6, 12, 18, 36, so a(36) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A000837, A001221, A018783, A056239, A289508, A289509, A296150, A298748, A318978, A327657.

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

a(n) = sumdiv(n, d, gcd(apply(x->primepi(x), factor(d)[,1])) == 1); \\ Michel Marcus, Jan 09 2019

a(n) = A000005(n) - A327657(n). - Antti Karttunen, Dec 05 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula