cp's OEIS Frontend

A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

%I A330991 #10 Feb 07 2021 06:25:46
%S A330991 4,6,8,9,10,14,15,16,21,22,24,25,26,27,30,32,33,34,35,38,39,40,42,46,
%T A330991 49,51,54,55,56,57,58,60,62,64,65,66,69,70,74,77,78,81,82,84,85,86,87,
%U A330991 88,90,91,93,94,95,96,102,104,105,106,110,111,114,115,118,119
%N A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).
%C A330991 In short, A001055(a(n)) belongs to A000040.
%H A330991 Amiram Eldar, <a href="/A330991/b330991.txt">Table of n, a(n) for n = 1..10000</a>
%H A330991 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 A330991 Factorizations of selected terms:
%e A330991   (4)    (8)      (16)       (24)       (60)       (96)
%e A330991   (2*2)  (2*4)    (2*8)      (3*8)      (2*30)     (2*48)
%e A330991          (2*2*2)  (4*4)      (4*6)      (3*20)     (3*32)
%e A330991                   (2*2*4)    (2*12)     (4*15)     (4*24)
%e A330991                   (2*2*2*2)  (2*2*6)    (5*12)     (6*16)
%e A330991                              (2*3*4)    (6*10)     (8*12)
%e A330991                              (2*2*2*3)  (2*5*6)    (2*6*8)
%e A330991                                         (3*4*5)    (3*4*8)
%e A330991                                         (2*2*15)   (4*4*6)
%e A330991                                         (2*3*10)   (2*2*24)
%e A330991                                         (2*2*3*5)  (2*3*16)
%e A330991                                                    (2*4*12)
%e A330991                                                    (2*2*3*8)
%e A330991                                                    (2*2*4*6)
%e A330991                                                    (2*3*4*4)
%e A330991                                                    (2*2*2*12)
%e A330991                                                    (2*2*2*2*6)
%e A330991                                                    (2*2*2*3*4)
%e A330991                                                    (2*2*2*2*2*3)
%t A330991 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A330991 Select[Range[100],PrimeQ[Length[facs[#]]]&]
%Y A330991 Factorizations are A001055, with image A045782, with complement A330976.
%Y A330991 Numbers whose number of strict integer partitions is prime are A035359.
%Y A330991 Numbers whose number of integer partitions is prime are A046063.
%Y A330991 Numbers whose number of set partitions is prime are A051130.
%Y A330991 Numbers whose number of factorizations is a power of 2 are A330977.
%Y A330991 The least number with prime(n) factorizations is A330992(n).
%Y A330991 Cf. A001222, A033833, A045783, A181819, A181821, A326622, A330972, A330973, A330993.
%K A330991 nonn
%O A330991 1,1
%A A330991 _Gus Wiseman_, Jan 07 2020