A033833 Highly factorable numbers: numbers with a record number of proper factorizations.

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

First differs from A045783 and A330972 in lacking 60.

Indices of records in A028422 or A001055.

Examples

			From _Gus Wiseman_, Jan 13 2020: (Start)
Factorizations of the initial terms:
  ()  (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.)
• E. R. Canfield, P. Erdős, C. Pomerance, On a problem of Oppenheim concerning Factorisatio Numerorum, J. Number Theory 17 (1983) 1-28, Table 1, column "n".
• Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.
• Arnold Knopfmacher and Michael Mays, Ordered and unordered factorizations of integers, Mathematica Journal, Vol. 10, No. 1 (2006), pp. 72-89.

Crossrefs

All terms belong to A025487 as well as to A330972.

The corresponding records are A272691.

The strict version is A331200.

Factorizations are A001055, with image A045782 and complement A330976.

Cf. A028422, A033834, A045778, A045783, A330973, A330998, A331049, A331050.

Programs

Mathematica

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

Formula

A001055(a(n)) = A272691(n). - Gus Wiseman, Jan 13 2020