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 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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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. LINKS Antti Karttunen, Table of n, a(n) for n = 1..20000 EXAMPLE The a(72) = 6 factorizations are (2*2*18), (2*6*6), (2*36), (4*18), (6*12), (72). 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}] PROG (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 CROSSREFS Positions of 0's are A318978. Cf. A000837, A001055, A007359, A085945, A289508, A289509, A302569, A302696, A320424. Sequence in context: A265330 A272216 A349906 * A321252 A225345 A335427 Adjacent sequences:  A320436 A320437 A320438 * A320440 A320441 A320442 KEYWORD nonn AUTHOR Gus Wiseman, Jan 08 2019 STATUS approved

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Last modified September 25 22:56 EDT 2022. Contains 356986 sequences. (Running on oeis4.)