cp's OEIS Frontend

A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

%I A330977 #5 Jan 08 2020 09:45:08
%S A330977 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 A330977 31,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,
%U A330977 62,63,65,67,68,69,71,72,73,74,75,76,77,79,82,83,85,86,87
%N A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.
%C A330977 The complement starts: 8, 16, 24, 27, 30, 32, 36, 40.
%H A330977 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 A330977 Factorizations of n = 1, 4, 12, 72:
%e A330977   ()  (4)    (12)     (72)
%e A330977       (2*2)  (2*6)    (8*9)
%e A330977              (3*4)    (2*36)
%e A330977              (2*2*3)  (3*24)
%e A330977                       (4*18)
%e A330977                       (6*12)
%e A330977                       (2*4*9)
%e A330977                       (2*6*6)
%e A330977                       (3*3*8)
%e A330977                       (3*4*6)
%e A330977                       (2*2*18)
%e A330977                       (2*3*12)
%e A330977                       (2*2*2*9)
%e A330977                       (2*2*3*6)
%e A330977                       (2*3*3*4)
%e A330977                       (2*2*2*3*3)
%t A330977 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A330977 Select[Range[100],IntegerQ[Log[2,Length[facs[#]]]]&]
%Y A330977 The same for strict integer partitions is A331022.
%Y A330977 Factorizations are A001055, with image A045782.
%Y A330977 The least number with exactly n factorizations is A330973(n).
%Y A330977 The least number with exactly 2^n factorizations is A330989(n).
%Y A330977 Numbers whose inverse prime shadow belongs to this sequence are A330990.
%Y A330977 Numbers with a prime number of factorizations are A330991.
%Y A330977 Cf. A002033, A033833, A045778, A045783, A317145, A326028, A330972, A330976.
%K A330977 nonn
%O A330977 1,2
%A A330977 _Gus Wiseman_, Jan 07 2020