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

1,1

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.

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.

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).

Example

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.

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]

Crossrefs

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

Sequence in context: A178428 A147760 A154148 * A242241 A257745 A153376

Adjacent sequences: A340815 A340816 A340817 * A340819 A340820 A340821

Keyword

nonn,frac

Author

Amiram Eldar, Jan 22 2021

Status

approved