A320439 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.

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

Offset: 1

Author

Gus Wiseman, Jan 08 2019

Keywords

• nonn

Comments

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.

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Positions of 0's are A318978.

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

Programs

Mathematica

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}]

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