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

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.

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

Examples

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

Links

• 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

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Extensions

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