A330977 - OEIS

%I #5 Jan 08 2020 09:45:08

%S 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,

%T 31,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,

%U 62,63,65,67,68,69,71,72,73,74,75,76,77,79,82,83,85,86,87

%N Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

%C The complement starts: 8, 16, 24, 27, 30, 32, 36, 40.

%H R. E. Canfield, P. Erdős and C. Pomerance, <a href="http://math.dartmouth.edu/~carlp/PDF/paper39.pdf">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</a>, J. Number Theory 17 (1983), 1-28.

%e Factorizations of n = 1, 4, 12, 72:

%e   ()  (4)    (12)     (72)

%e       (2*2)  (2*6)    (8*9)

%e              (3*4)    (2*36)

%e              (2*2*3)  (3*24)

%e                       (4*18)

%e                       (6*12)

%e                       (2*4*9)

%e                       (2*6*6)

%e                       (3*3*8)

%e                       (3*4*6)

%e                       (2*2*18)

%e                       (2*3*12)

%e                       (2*2*2*9)

%e                       (2*2*3*6)

%e                       (2*3*3*4)

%e                       (2*2*2*3*3)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Select[Range[100],IntegerQ[Log[2,Length[facs[#]]]]&]

%Y The same for strict integer partitions is A331022.

%Y Factorizations are A001055, with image A045782.

%Y The least number with exactly n factorizations is A330973(n).

%Y The least number with exactly 2^n factorizations is A330989(n).

%Y Numbers whose inverse prime shadow belongs to this sequence are A330990.

%Y Numbers with a prime number of factorizations are A330991.

%Y Cf. A002033, A033833, A045778, A045783, A317145, A326028, A330972, A330976.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 07 2020