A088533 Numbers k such that bigomega(k!)/omega(k!) is an integer.

2, 3, 4, 7, 15, 22, 24, 40, 49, 58, 71, 74, 92, 124, 179, 183, 232, 237, 413, 542, 547, 731, 752, 758, 983, 1266, 1283, 1289, 1336, 1706, 1712, 1725, 2656, 2909, 3509, 3612, 3653, 3674, 3702, 3709, 4617, 4646, 4697, 5993

Offset: 1

Author

Cino Hilliard, Nov 16 2003

Keywords

• nonn

Examples

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

Links

• Ivan Neretin, David A. Corneth, and Charles R Greathouse IV, Table of n, a(n) for n = 1..344 (1..208 from Neretin, 209..266 from Corneth, 267..344 from Greathouse)
• M. Hassani, On the decomposition of n! into primes, arXiv:math/0606316 [math.NT], 2006-2007.

Crossrefs

Cf. A000720, A001221, A001222, A022559.

Programs

Mathematica

ointQ[n_]:=Module[{f=n!},IntegerQ[PrimeOmega[f]/PrimeNu[f]]]; Select[Range[ 2,6000],ointQ] (* Harvey P. Dale, Dec 07 2013 *)
Omega = Nu = 0; a = {}; Do[If[PrimeQ[n], Nu++]; Omega += PrimeOmega[n];
If[Divisible[Omega, Nu], AppendTo[a, n]], {n, 2, 6000}]; a (* Ivan Neretin, Mar 14 2017 *)

PARI

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

PARI

is(n)=my(s); forprime(p=2,n, my(k=n\p); while(k, s+=k; k\=p)); s%primepi(n)==0 \\ Charles R Greathouse IV, Feb 28 2025

PARI

list(lim)=my(v=List(),b,s); forfactored(n=2,lim\1, b+=bigomega(n); if(n[2][,2]==[1]~, s++); if(b%s==0, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Feb 28 2025

Formula

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.