A327405 - OEIS

A327405

Quotient of n over the maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 32, 3, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 2, 5, 8, 1, 2, 1, 12, 1, 2, 1, 64, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5

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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 whose prime indices have a common divisor > 1 are listed in A318978.

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

FORMULA

a(n) = n/A327656(n).

EXAMPLE

The divisors of 90 that are 1 or whose prime indices have a common divisor > 1 are {1, 3, 5, 9}, so a(90) = 90/9 = 10.

MATHEMATICA

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

PROG

(PARI) A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi, factor(d)[, 1]~))>1), divisors(n)))); \\ Antti Karttunen, Dec 06 2021