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A327525 Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime. 0
1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 1, 2, 7, 1, 1, 4, 2, 1, 12, 2, 4, 1, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 1, 2, 4, 2, 1, 12, 2, 1, 2, 2, 7, 1, 4, 2, 2, 2, 19, 1, 1, 5, 1, 7, 2, 1, 1, 5, 12, 1, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..85.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

FORMULA

a(n) = A001055(A302569(n)).

EXAMPLE

The a(47) = 11 factorizations of 60 together with the corresponding multiset partitions of {1,1,2,3}:

  (2*2*3*5)  {{1},{1},{2},{3}}

  (2*2*15)   {{1},{1},{2,3}}

  (2*3*10)   {{1},{2},{1,3}}

  (2*5*6)    {{1},{3},{1,2}}

  (2*30)     {{1},{1,2,3}}

  (3*4*5)    {{2},{1,1},{3}}

  (3*20)     {{2},{1,1,3}}

  (4*15)     {{1,1},{2,3}}

  (5*12)     {{3},{1,1,2}}

  (6*10)     {{1,2},{1,3}}

  (60)       {{1,1,2,3}}

MATHEMATICA

nn=100;

facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];

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

y=Select[Range[nn], PrimeQ[#]||CoprimeQ@@primeMS[#]&];

Table[Length[facsusing[Rest[y], n]], {n, y}]

CROSSREFS

See link for additional cross-references.

Cf. A056239, A112798, A281116, A318721, A302569, A304711, A305079.

Sequence in context: A134583 A087467 A231568 * A327540 A227687 A128118

Adjacent sequences:  A327522 A327523 A327524 * A327526 A327527 A327528

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 20 2019

STATUS

approved

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Last modified July 28 19:33 EDT 2021. Contains 346335 sequences. (Running on oeis4.)