A327512 -id:A327512 - 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).

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, 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, 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

Offset: 1

Gus Wiseman, Sep 19 2019

nonn

Positions of zeros are A289509.
First term > 1 is a(225) = 2.
First zero not in A318978 is a(17719) = 0.
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.

			We have 50625 -> 3375 -> 225 ->  15 -> 1, so a(50625) = 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 related to prime indices in the factorization of n.

See link for additional cross-references.

Cf. A000005, A006530, A056239, A112798, A289509, A302569, A302696, A304711.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[FixedPointList[#/Max[Select[Divisors[#],#==1||CoprimeQ@@primeMS[#]&]]&,n]]-2,{n,100}]

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))));
A327512(n) = vecmax(select(isA302696, divisors(n)));
A327515(n) = for(k=0,oo,my(nextn=n/A327512(n)); if(nextn==n,return(k)); n = nextn); \\ Antti Karttunen, Jan 28 2025

a(15^n) = n.

Data section extended to a(105) and secondary offset added by Antti Karttunen, Jan 28 2025

A302696 Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 38, 40, 44, 46, 48, 51, 52, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 74, 76, 77, 80, 82, 85, 86, 88, 92, 93, 94, 95, 96, 102, 104, 106, 110, 112, 116, 118, 119, 120, 122, 123, 124, 128, 132

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair has a common divisor other than 1. A single number is not considered coprime unless it is equal to 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Number 36 = prime(1)*prime(1)*prime(2)*prime(2) is not included in the sequence, because the pair of prime indices {2,2} is not coprime. - Gus Wiseman, Dec 06 2021

			Sequence of integer partitions with pairwise coprime parts begins: (), (1), (11), (21), (111), (31), (211), (41), (32), (1111), (311), (51), (2111), (61), (411), (321).
Missing from this list are: (2), (3), (4), (22), (5), (6), (7), (221), (8), (42), (9), (33), (222).

Cf. A000837, A000961, A001222, A005117, A007359, A051424, A275024, A289508, A289509, A298748, A302568, A302569, A302697, A302698, A327512, A327513.

filter:= proc(n) local F;
   F:= ifactors(n)[2];
   if nops(F)=1 then if F[1][1] = 2 then return true else return false fi fi;
   if ormap(t -> t[2]>1 and t[1] <> 2, F) then return false fi;
   F:= map(t -> numtheory:-pi(t[1]), F);
   ilcm(op(F))=convert(F,`*`)
end proc:
select(filter, [$1..200]); # Robert Israel, Sep 10 2020

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Or[#===1,CoprimeQ@@primeMS[#]]&]

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)))); \\ Antti Karttunen, Dec 06 2021

Clarification (with repetition) added to the definition by Antti Karttunen, Dec 06 2021

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).

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