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%I #14 Dec 06 2021 11:03:13
%S 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,
%T 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,
%U 1,2,0,6,0,1,1,2,1,1,0,7,0,1,0,3,1,1,0
%N Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.
%C Also the number of multiset partitions of the multiset of prime indices of n using multisets each of which is relatively prime.
%C 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.
%C 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.
%H Antti Karttunen, <a href="/A320439/b320439.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e The a(72) = 6 factorizations are (2*2*18), (2*6*6), (2*36), (4*18), (6*12), (72).
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t facsrp[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[facsrp[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],GCD@@primeMS[#]==1&]}]];
%t Table[Length[facsrp[n]],{n,100}]
%o (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
%Y Positions of 0's are A318978.
%Y Cf. A000837, A001055, A007359, A085945, A289508, A289509, A302569, A302696, A320424.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 08 2019
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