cp's OEIS Frontend

A088530 Denominator of bigomega(n)/omega(n).

%I A088530 #36 Jun 27 2025 09:21:15
%S A088530 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,
%T A088530 1,1,1,1,1,1,1,1,2,2,1,1,2,1,2,1,2,1,1,1,1,1,1,1,3,1,1,2,1,1,1,1,2,1,
%U A088530 1,1,2,1,1,2,2,1,1,1,2,1,1,1,3,1,1,1,1,1,3,1,2,1,1,1,1,1,2,2,1
%N A088530 Denominator of bigomega(n)/omega(n).
%C A088530 a(n) is the denominator of A022559(n)/A000720(n). - _Robert Israel_, Jan 08 2024
%H A088530 Antti Karttunen, <a href="/A088530/b088530.txt">Table of n, a(n) for n = 2..10000</a>
%H A088530 R. L. Duncan, <a href="https://doi.org/10.1090/S0002-9939-1970-0252311-1">On the factorization of integers</a>, Proc. Amer. Math. Soc. 25 (1970), 191-192.
%H A088530 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A088530 Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = denominator of B/O.
%e A088530 bigomega(24) / omega(24) = 4/2 = 2/1, so a(24) = 1.
%p A088530 N:= 100:
%p A088530 W:= ListTools:-PartialSums(map(numtheory:-bigomega,[$1..N])):
%p A088530 seq(denom(W[i]/numtheory:-pi(i)),i=2..N); # _Robert Israel_, Jan 08 2024
%t A088530 Table[Denominator[PrimeOmega[n]/PrimeNu[n]],{n,2,100}] (* _Harvey P. Dale_, Mar 22 2012 *)
%o A088530 (PARI) for(x=2,100,y=bigomega(x)/omega(x);print1(denominator(y)","))
%o A088530 (Python)
%o A088530 from sympy import primefactors, Integer
%o A088530 def bigomega(n): return 0 if n==1 else bigomega(Integer(n)/primefactors(n)[0]) + 1
%o A088530 def omega(n): return Integer(len(primefactors(n)))
%o A088530 def a(n): return (bigomega(n)/omega(n)).denominator
%o A088530 print([a(n) for n in range(2, 51)]) # _Indranil Ghosh_, Jul 13 2017
%Y A088530 Cf. A001221, A001222, A000720, A022559, A070012, A070013, A070014, A088529 (gives the numerator).
%K A088530 nonn,frac,look
%O A088530 2,11
%A A088530 _Cino Hilliard_, Nov 16 2003