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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. 0
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

Table of n, a(n) for n=1..87.

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}]

CROSSREFS

Positions of 0's are A318978.

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

Sequence in context: A007814 A265330 A272216 * A321252 A225345 A083280

Adjacent sequences:  A320436 A320437 A320438 * A320440 A320441 A320442

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 08 2019

STATUS

approved

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Last modified February 26 08:15 EST 2020. Contains 332277 sequences. (Running on oeis4.)