A327538 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, prime, or whose prime indices are relatively prime (A327535, A327537).

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2

Offset: 1

Author

Gus Wiseman, Sep 17 2019

Keywords

• nonn

Comments

The first index m such that a(m) > 1 but m is not in A322336 is m = 2335.

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 that are 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

Examples

			We have 441 -> 63 -> 9 -> 3 -> 1, so a(441) = 4.
		

Links

• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

See link for additional cross-references.

Cf. A000005, A001222, A006530, A056239, A112798, A289509, A327407.

Programs

Mathematica

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

Formula

a(1) = 0; if n is prime or has relatively prime prime indices, then a(n) = 1; otherwise a(n) = Omega(n) = A001222(n).