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<a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

Antti Karttunen, <a href="/A320439/b320439.txt">Table of n, a(n) for n = 1..20000</a>

(PARI) A320439(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d<=m)&&(1==gcd(apply(x->primepi(x), factor(d)[, 1]))), s += A320439(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2021

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Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Factorizations Number of factorizations of n using elements of A289509.

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allocated for Gus WisemanNumber of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Factorizations of n using elements of A289509.

1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 5, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 2, 0, 7, 1, 1, 1, 3, 0, 1, 0, 4, 0, 1, 0, 2, 1, 1, 0, 7, 0, 1, 1, 2, 0, 1, 1, 4, 0, 1, 0, 5, 0, 1, 0, 11, 0, 2, 0, 2, 1, 2, 0, 6, 0, 1, 1, 2, 1, 1, 0, 7, 0, 1, 0, 3, 1, 1, 0

1,4

Also the number of multiset partitions of the multiset of prime indices of n using multisets each of which is relatively prime.

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.

Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

The a(72) = 6 factorizations are (2*2*18), (2*6*6), (2*36), (4*18), (6*12), (72).

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

facsrp[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#, d]&)/@Select[facsrp[n/d], Min@@#>=d&], {d, Select[Rest[Divisors[n]], GCD@@primeMS[#]==1&]}]];

Table[Length[facsrp[n]], {n, 100}]

Positions of 0's are A318978.

Cf. A000837, A001055, A007359, A085945, A289508, A289509, A302569, A302696, A320424.

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Gus Wiseman, Jan 08 2019

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