cp's OEIS Frontend

A327515 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, 2, or a nonprime number whose prime indices are pairwise coprime (A327512, A327514).

%I A327515 #11 Jan 28 2025 20:51:29
%S A327515 0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,
%T A327515 1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,
%U A327515 1,1,0,1,0,1,1,1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,1,1,0,1,1,1,0,1,0,1,1
%N A327515 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, 2, or a nonprime number whose prime indices are pairwise coprime (A327512, A327514).
%C A327515 Positions of zeros are A289509.
%C A327515 First term > 1 is a(225) = 2.
%C A327515 First zero not in A318978 is a(17719) = 0.
%C A327515 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, 2, or a nonprime number whose prime indices are pairwise coprime are listed in A302696.
%H A327515 Antti Karttunen, <a href="/A327515/b327515.txt">Table of n, a(n) for n = 1..65537</a>
%H A327515 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 A327515 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A327515 a(15^n) = n.
%e A327515 We have 50625 -> 3375 -> 225 ->  15 -> 1, so a(50625) = 4.
%t A327515 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A327515 Table[Length[FixedPointList[#/Max[Select[Divisors[#],#==1||CoprimeQ@@primeMS[#]&]]&,n]]-2,{n,100}]
%o A327515 (PARI)
%o A327515 isA302696(n) = if(isprimepower(n), !(n%2), if(!issquarefree(n>>valuation(n, 2)), 0, my(pis=apply(primepi, factor(n)[, 1])); (lcm(pis)==factorback(pis))));
%o A327515 A327512(n) = vecmax(select(isA302696, divisors(n)));
%o A327515 A327515(n) = for(k=0,oo,my(nextn=n/A327512(n)); if(nextn==n,return(k)); n = nextn); \\ _Antti Karttunen_, Jan 28 2025
%Y A327515 See link for additional cross-references.
%Y A327515 Cf. A000005, A006530, A056239, A112798, A289509, A302569, A302696, A304711.
%K A327515 nonn
%O A327515 1,225
%A A327515 _Gus Wiseman_, Sep 19 2019
%E A327515 Data section extended to a(105) and secondary offset added by _Antti Karttunen_, Jan 28 2025