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

1,2

Comments

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

Links

Table of n, a(n) for n=1..67.

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

Example

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)

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[#]]]]&]

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.

Sequence in context: A078129 A325368 A325251 * A188437 A344414 A356438

Adjacent sequences: A330974 A330975 A330976 * A330978 A330979 A330980

Keyword

nonn

Author

Gus Wiseman, Jan 07 2020

Status

approved