cp's OEIS Frontend

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

%I A088533 #30 Mar 03 2025 21:39:18
%S A088533 2,3,4,7,15,22,24,40,49,58,71,74,92,124,179,183,232,237,413,542,547,
%T A088533 731,752,758,983,1266,1283,1289,1336,1706,1712,1725,2656,2909,3509,
%U A088533 3612,3653,3674,3702,3709,4617,4646,4697,5993
%N A088533 Numbers k such that bigomega(k!)/omega(k!) is an integer.
%H A088533 Ivan Neretin, David A. Corneth, and Charles R Greathouse IV, <a href="/A088533/b088533.txt">Table of n, a(n) for n = 1..344</a> (1..208 from Neretin, 209..266 from Corneth, 267..344 from Greathouse)
%H A088533 M. Hassani, <a href="http://arxiv.org/abs/math/0606316">On the decomposition of n! into primes</a>, arXiv:math/0606316 [math.NT], 2006-2007.
%F A088533 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 A088533 S(4!) = bigomega(4!) / omega(4!) = 4/2 = 2 so 4 is 3rd term in the sequence.
%t A088533 ointQ[n_]:=Module[{f=n!},IntegerQ[PrimeOmega[f]/PrimeNu[f]]]; Select[Range[ 2,6000],ointQ] (* _Harvey P. Dale_, Dec 07 2013 *)
%t A088533 Omega = Nu = 0; a = {}; Do[If[PrimeQ[n], Nu++]; Omega += PrimeOmega[n];
%t A088533 If[Divisible[Omega, Nu], AppendTo[a, n]], {n, 2, 6000}]; a (* _Ivan Neretin_, Mar 14 2017 *)
%o A088533 (PARI) for(x=2,10000,x1=x!;y=bigomega(x1)/omega(x1);if(y==floor(y),print1((x)",")))
%o A088533 (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
%o A088533 (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
%Y A088533 Cf. A000720, A001221, A001222, A022559.
%K A088533 nonn
%O A088533 1,1
%A A088533 _Cino Hilliard_, Nov 16 2003