A327525 Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.

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

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