A317176 Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 4, 1, 18, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 49, 1, 3, 1, 3, 1, 11, 1, 11, 1, 1, 1, 21, 1, 1, 3, 74, 1, 4, 1, 3, 1, 4, 1, 78, 1, 1, 3, 3, 1, 4, 1, 49, 6, 1, 1

Offset

1,8

Comments

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

Links

Table of n, a(n) for n=1..83.

Formula

a(prime^n) = A213427(n).

Example

The a(24) = 11 chains:
  (2*2*2*3) < (24)
  (2*2*2*3) < (2*12)  < (24)
  (2*2*2*3) < (3*8)   < (24)
  (2*2*2*3) < (4*6)   < (24)
  (2*2*2*3) < (2*2*6) < (24)
  (2*2*2*3) < (2*3*4) < (24)
  (2*2*2*3) < (2*2*6) < (2*12) < (24)
  (2*2*2*3) < (2*2*6) < (4*6)  < (24)
  (2*2*2*3) < (2*3*4) < (2*12) < (24)
  (2*2*2*3) < (2*3*4) < (3*8)  < (24)
  (2*2*2*3) < (2*3*4) < (4*6)  < (24)

Crossrefs

Cf. A001055, A002846, A007716, A045778, A162247, A213427, A275024, A281113, A299202, A317144, A317145, A317146.

Sequence in context: A316767 A292505 A281119 * A318812 A337066 A324247

Adjacent sequences: A317173 A317174 A317175 * A317177 A317178 A317179

Keyword

nonn

Author

Gus Wiseman, Jul 23 2018

Status

approved