A272691 Number of factorizations of the highly factorable numbers A033833.

1, 2, 3, 4, 5, 7, 9, 12, 16, 19, 21, 29, 30, 31, 38, 47, 52, 57, 64, 77, 98, 105, 109, 118, 171, 212, 289, 382, 392, 467, 484, 662, 719, 737, 783, 843, 907, 1097, 1261, 1386, 1397, 1713, 1768, 2116, 2179, 2343, 3079, 3444, 3681, 3930, 5288, 5413, 5447

Offset

1,2

Comments

These are defined as record numbers of factorizations (A001055). - Gus Wiseman, Jan 13 2020

Links

Amiram Eldar, Table of n, a(n) for n = 1..235 (terms 1..118 from E. R. Canfield et al.)

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Formula

a(n) = A001055(A033833(n)).

a(n) = A033834(n) + 1. - Amiram Eldar, Jun 07 2019

Example

From Gus Wiseman, Jan 13 2020: (Start)
The a(1) = 1 through a(8) = 12 factorizations of highly factorable numbers:
  ()  (4)    (8)      (12)     (16)       (24)       (36)       (48)
      (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (4*9)      (6*8)
             (2*2*2)  (3*4)    (4*4)      (4*6)      (6*6)      (2*24)
                      (2*2*3)  (2*2*4)    (2*12)     (2*18)     (3*16)
                               (2*2*2*2)  (2*2*6)    (3*12)     (4*12)
                                          (2*3*4)    (2*2*9)    (2*3*8)
                                          (2*2*2*3)  (2*3*6)    (2*4*6)
                                                     (3*3*4)    (3*4*4)
                                                     (2*2*3*3)  (2*2*12)
                                                                (2*2*2*6)
                                                                (2*2*3*4)
                                                                (2*2*2*2*3)
(End)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[facs[n]], {n, 100}]//.{foe___, x_, y_, afe___}/; x>=y:>{foe, x, afe} (* Gus Wiseman, Jan 13 2020 *)

Crossrefs

The strict version is A331232.

Factorizations are A001055, with image A045782 and complement A330976.

Highly factorable numbers are A033833, with strict version A331200.

Cf. A033834, A045778, A045783, A330972, A330973, A330998.

Sequence in context: A039861 A039877 A291833 * A192433 A189083 A318156

Adjacent sequences: A272688 A272689 A272690 * A272692 A272693 A272694

Keyword

nonn

Author

N. J. A. Sloane, Jun 02 2016, following a suggestion from George Beck.

Status

approved