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).

%I A327406 #15 Dec 07 2021 11:07:28
%S A327406 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,
%T A327406 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,
%U A327406 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
%N 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).
%C A327406 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.
%C A327406 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
%H A327406 Antti Karttunen, <a href="/A327406/b327406.txt">Table of n, a(n) for n = 1..65537</a>
%H A327406 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>
%H A327406 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A327406 We have 5115 -> 165 -> 15 -> 3 -> 1, so a(5115) = 4.
%t A327406 Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]]&,n]]-2,{n,100}]
%o A327406 (PARI)
%o A327406 A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))));
%o A327406 A327406(n) = { my(u = A327405(n), k=0); while(u!=n, k++; n = u; u = A327405(n)); (k); }; \\ _Antti Karttunen_, Dec 06 2021
%Y A327406 First appearance of n is A080696(n).
%Y A327406 See link for additional cross-references.
%Y A327406 Cf. A000005, A000079 (positions of 0's), A056239, A112798, A281116, A289509, A302569, A318978.
%K A327406 nonn
%O A327406 1,15
%A A327406 _Gus Wiseman_, Sep 21 2019
%E A327406 Data section extended up to 105 terms by _Antti Karttunen_, Dec 06 2021