A327405 -id:A327405 - cp's OEIS frontend

A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2

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.

Note that A318978 includes also all odd primes and their powers, thus the only numbers for which a maximum such divisor is 1 are the powers of 2. Therefore A000079 gives the indices of zeros in this sequence. - Antti Karttunen, Dec 06 2021

			We have 5115 -> 165 -> 15 -> 3 -> 1, so a(5115) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

First appearance of n is A080696(n).
See link for additional cross-references.
Cf. A000005, A000079 (positions of 0's), A056239, A112798, A281116, A289509, A302569, A318978.

Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]]&,n]]-2,{n,100}]

A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))));
A327406(n) = { my(u = A327405(n), k=0); while(u!=n, k++; n = u; u = A327405(n)); (k); }; \\ Antti Karttunen, Dec 06 2021

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

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

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