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A330977
Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.
13
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
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.
Sequence in context: A078129 A325368 A325251 * A188437 A344414 A356438
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 07 2020
STATUS
approved