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