A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Author

Gus Wiseman, Jan 12 2020

Keywords

• nonn

Comments

First differs from A080257 in lacking 60.

Examples

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)
		

Crossrefs

The version for strict integer partitions is A035359.

The version for integer partitions is A046063.

The version for set partitions is A051130.

The non-strict version is A330991.

Factorizations are A001055 with image A045782 and complement A330976.

Strict factorizations are A045778 with image A045779 and complement A330975.

Numbers whose number of strict factorizations is odd are A331230.

Numbers whose number of strict factorizations is even are A331231.

The least number with n strict factorizations is A330974(n).

Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

Programs

Mathematica

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