%I #15 Jul 20 2023 11:39:20

%S 2,3,4,7,15,22,24,40,49,58,71,74,92,124,179,183,232,237,413,542,547,

%T 731,752,758,983,1266,1283,1289,1336,1706,1712,1725,2656,2909,3509,

%U 3612,3653,3674,3702,3709,4617,4646,4697,5993

%N Numbers n such that Bigomega(n!)/Omega(n!) is an integer.

%H Ivan Neretin, <a href="/A088533/b088533.txt">Table of n, a(n) for n = 1..208</a>

%H M. Hassani, <a href="http://arxiv.org/abs/math/0606316">On the decomposition of n! into primes</a>, arXiv:math/0606316.

%F Let k = number of prime divisors of n! counted with multiplicity; b = number of distinct prime divisors of n!. Then n is in sequence if k/b is an integer.

%e S(4!) = bigomega(4!) / omega(4!) = 4/2 = 2 so 4 is 3rd term in the sequence.

%t ointQ[n_]:=Module[{f=n!},IntegerQ[PrimeOmega[f]/PrimeNu[f]]]; Select[Range[ 2,6000],ointQ] (* _Harvey P. Dale_, Dec 07 2013 *)

%t Omega = Nu = 0; a = {}; Do[If[PrimeQ[n], Nu++]; Omega += PrimeOmega[n];

%t If[Divisible[Omega, Nu], AppendTo[a, n]], {n, 2, 6000}]; a (* _Ivan Neretin_, Mar 14 2017 *)

%o (PARI) for(x=2,10000,x1=x!;y=bigomega(x1)/omega(x1);if(y==floor(y),print1((x)",")))

%Y Cf. A000720, A001221, A001222, A022559.

%K nonn

%O 1,1

%A _Cino Hilliard_, Nov 16 2003