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%I #36 Dec 15 2022 13:35:44
%S 1,1,2,1,1,1,3,2,1,1,3,1,1,1,4,1,3,1,3,1,1,1,2,2,1,3,3,1,1,1,5,1,1,1,
%T 2,1,1,1,2,1,1,1,3,3,1,1,5,2,3,1,3,1,2,1,2,1,1,1,4,1,1,3,6,1,1,1,3,1,
%U 1,1,5,1,1,3,3,1,1,1,5,4,1,1,4,1,1,1,2,1,4,1,3,1,1,1,3,1,3,3,2
%N Numerator of Bigomega(n)/Omega(n).
%D H. Z. Cao, On the average of exponents, Northeast. Math. J., Vol. 10 (1994), pp. 291-296.
%H Antti Karttunen, <a href="/A088529/b088529.txt">Table of n, a(n) for n = 2..10000</a>
%H R. L. Duncan, <a href="https://doi.org/10.1090/S0002-9939-1970-0252311-1">On the factorization of integrers</a>, Proc. Amer. Math. Soc. 25 (1970), 191-192.
%H Steven Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022.
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = numerator of B/O.
%F a(n) = A136565(n) = A181591(n) for n: 2 <= n < 24. - _Reinhard Zumkeller_, Nov 01 2010
%F Sum_{k=2..n} a(k)/A088530(k) ~ n + O(n/log(log(n))) (Duncan, 1970). - _Amiram Eldar_, Oct 14 2022
%F Sum_{k=2..n} a(k)/A088530(k) = n + c_1 * n/log(log(n)) + c_2 * n/log(log(n))^2 + O(n/log(log(n))^3), where c_1 = A136141 and c_2 = A272531 (Cao, 1994; Finch, 2020). - _Amiram Eldar_, Dec 15 2022
%e bigomega(24) / omega(24) = 4/2 = 2, so a(24) = 2.
%t Table[Numerator[PrimeOmega[n]/PrimeNu[n]], {n, 2, 100}] (* _Michael De Vlieger_, Jul 12 2017 *)
%o (PARI) for(x=2,100,y=bigomega(x)/omega(x);print1(numerator(y)","))
%o (Python)
%o from sympy import primefactors, Integer
%o def bigomega(n): return 0 if n==1 else bigomega(Integer(n)/primefactors(n)[0]) + 1
%o def omega(n): return Integer(len(primefactors(n)))
%o def a(n): return (bigomega(n)/omega(n)).numerator()
%o print([a(n) for n in range(2, 51)]) # _Indranil Ghosh_, Jul 13 2017
%Y Cf. A001221, A001222, A070012, A070013, A070014, A088530 (gives the denominator).
%Y Cf. A136141, A272531.
%K nonn,frac
%O 2,3
%A _Cino Hilliard_, Nov 16 2003
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