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

Author

Gus Wiseman, Jul 23 2018

Keywords

• nonn

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.

Examples

			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.

Formula

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