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

Author

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

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

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.

Programs

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 *)

Formula

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

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