A340818 Numerators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

5, 7, 41, 3, 197, 229, 5827, 277, 1157, 8382, 268049, 94175911, 964941119, 1929224113, 31529606831, 835346466959, 3398377571053, 52665885581009, 119955940157647877, 34063199364211668943, 315047077264055066629, 199089493729235251718903, 47411489829747180146759

Offset: 1

Author

Amiram Eldar, Jan 22 2021

Keywords

• nonn
• frac

Comments

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).

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.

Examples

			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, ...
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..462
• Jérémie Detrey, Pierre-Jean Spaenlehauer and Paul Zimmermann, Computing the rho constant, 2016.
• Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Crossrefs

Cf. A005408, A036554, A046660, A162644, A340819 (denominators), A340820.

Programs

Mathematica

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]

Formula

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).

Limit_{n->oo} f(n) = 0.73584... (A340820).