A327540 Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 7, 1, 4, 1, 4, 3, 2, 1, 12, 3, 2, 4, 1, 4, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 2, 1, 9, 2, 2, 7, 1, 9, 4, 2, 2, 2, 19, 1

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. Numbers that are 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).

Links

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

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Example

The a(74) = 9 factorizations of 84 together with the corresponding multiset partitions of {1,1,2,4}:
  (2*2*3*7)  {{1},{1},{2},{4}}
  (2*3*14)   {{1},{2},{1,4}}
  (2*6*7)    {{1},{1,2},{4}}
  (2*42)     {{1},{1,2,4}}
  (3*4*7)    {{2},{1,1},{4}}
  (3*28)     {{2},{1,1,4}}
  (6*14)     {{1,2},{1,4}}
  (7*12)     {{4},{1,1,2}}
  (84)       {{1,1,2,4}}

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, #]&]}]];
y=Select[Range[nn], #==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
Table[Length[facsusing[Rest[y], n]], {n, y}]

Crossrefs

See link for additional cross-references.

Cf. A006530, A056239, A112798, A281116, A289509, A327407.

Sequence in context: A087467 A231568 A327525 * A227687 A128118 A205696

Adjacent sequences: A327537 A327538 A327539 * A327541 A327542 A327543

Keyword

nonn

Author

Gus Wiseman, Sep 17 2019

Status

approved