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

1,1

Comments

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

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.

Example

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)

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[#]]]&]

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.

Sequence in context: A132858 A071941 A188654 * A180366 A340656 A346041

Adjacent sequences: A330988 A330989 A330990 * A330992 A330993 A330994

Keyword

nonn

Author

Gus Wiseman, Jan 07 2020

Status

approved