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%I #12 Jan 28 2021 03:35:57
%S 5,7,41,3,197,229,5827,277,1157,8382,268049,94175911,964941119,
%T 1929224113,31529606831,835346466959,3398377571053,52665885581009,
%U 119955940157647877,34063199364211668943,315047077264055066629,199089493729235251718903,47411489829747180146759
%N Numerators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).
%C Let Omega_n(k) be the number of prime divisors of k not exceeding prime(n) counted with multiplicity, and omega_n(k) the number of distinct prime divisors of k not exceeding prime(n). Then, f(n) = a(n)/A340819(n) is the asymptotic density of numbers k such that Omega_n(k) == omega_n(k) (mod 2).
%C Equivalently, f(n) is the asymptotic density of numbers k such that A046660(d_n(k)) is even, where d_n(k) is the largest prime(n)-smooth divisor of k.
%H Amiram Eldar, <a href="/A340818/b340818.txt">Table of n, a(n) for n = 1..462</a>
%H Jérémie Detrey, Pierre-Jean Spaenlehauer and Paul Zimmermann, <a href="https://members.loria.fr/PZimmermann/papers/rho.pdf">Computing the rho constant</a>, 2016.
%H Michael J. Mossinghoff and Timothy S. Trudgian, <a href="https://arxiv.org/abs/1906.02847">A tale of two omegas</a>, arXiv:1906.02847 [math.NT], 2019.
%F Let delta(n) = 1/(prime(n)*(prime(n)+1)) be the asymptotic density of numbers whose prime(n)-adic valuation is positive and even. Let f(0) = 1. Then, f(n) = f(n-1)*(1 - delta(n)) + (1 - f(n))*delta(n).
%F Limit_{n->oo} f(n) = 0.73584... (A340820).
%e The sequence of fractions begins with 5/6, 7/9, 41/54, 3/4, 197/264, 229/308, 5827/7854, 277/374, 1157/1564, 8382/11339, ...
%e For n=1, Omega_2(k)-omega_2(k) is even for either odd k (A005408), or even k whose binary representation ends in an odd number of zeros (A036554). The disjoint union of these 2 sequences has an asymptotic density 1/2 + 1/3 = 5/6.
%t d[p_] := 1/(p*(p + 1)); delta[n_] := delta[n] = d[Prime[n]]; f[0] = 1; f[n_] := f[n] = f[n - 1] * (1 - delta[n]) + (1 - f[n - 1]) * delta[n]; Numerator @ Array[f, 30]
%Y Cf. A005408, A036554, A046660, A162644, A340819 (denominators), A340820.
%K nonn,frac
%O 1,1
%A _Amiram Eldar_, Jan 22 2021