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A330991
Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).
17
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
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).
Sequence in context: A132858 A071941 A188654 * A180366 A340656 A373482
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 07 2020
STATUS
approved