A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119

Offset: 1

Author

Gus Wiseman, Jan 07 2020

Keywords

• nonn

Comments

In short, A001055(a(n)) belongs to A000040.

Examples

			Factorizations of selected terms:
  (4)    (8)      (16)       (24)       (60)       (96)
  (2*2)  (2*4)    (2*8)      (3*8)      (2*30)     (2*48)
         (2*2*2)  (4*4)      (4*6)      (3*20)     (3*32)
                  (2*2*4)    (2*12)     (4*15)     (4*24)
                  (2*2*2*2)  (2*2*6)    (5*12)     (6*16)
                             (2*3*4)    (6*10)     (8*12)
                             (2*2*2*3)  (2*5*6)    (2*6*8)
                                        (3*4*5)    (3*4*8)
                                        (2*2*15)   (4*4*6)
                                        (2*3*10)   (2*2*24)
                                        (2*2*3*5)  (2*3*16)
                                                   (2*4*12)
                                                   (2*2*3*8)
                                                   (2*2*4*6)
                                                   (2*3*4*4)
                                                   (2*2*2*12)
                                                   (2*2*2*2*6)
                                                   (2*2*2*3*4)
                                                   (2*2*2*2*2*3)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Crossrefs

Factorizations are A001055, with image A045782, with complement A330976.

Numbers whose number of strict integer partitions is prime are A035359.

Numbers whose number of integer partitions is prime are A046063.

Numbers whose number of set partitions is prime are A051130.

Numbers whose number of factorizations is a power of 2 are A330977.

The least number with prime(n) factorizations is A330992(n).

Cf. A001222, A033833, A045783, A181819, A181821, A326622, A330972, A330973, A330993.

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],PrimeQ[Length[facs[#]]]&]