A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87

Offset: 1

Author

Gus Wiseman, Jan 07 2020

Keywords

• nonn

Comments

The complement starts: 8, 16, 24, 27, 30, 32, 36, 40.

Examples

			Factorizations of n = 1, 4, 12, 72:
  ()  (4)    (12)     (72)
      (2*2)  (2*6)    (8*9)
             (3*4)    (2*36)
             (2*2*3)  (3*24)
                      (4*18)
                      (6*12)
                      (2*4*9)
                      (2*6*6)
                      (3*3*8)
                      (3*4*6)
                      (2*2*18)
                      (2*3*12)
                      (2*2*2*9)
                      (2*2*3*6)
                      (2*3*3*4)
                      (2*2*2*3*3)
		

Links

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

Crossrefs

The same for strict integer partitions is A331022.

Factorizations are A001055, with image A045782.

The least number with exactly n factorizations is A330973(n).

The least number with exactly 2^n factorizations is A330989(n).

Numbers whose inverse prime shadow belongs to this sequence are A330990.

Numbers with a prime number of factorizations are A330991.

Cf. A002033, A033833, A045778, A045783, A317145, A326028, A330972, A330976.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],IntegerQ[Log[2,Length[facs[#]]]]&]