A340819 -id:A340819 - cp's OEIS frontend

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

Amiram Eldar, Jan 22 2021

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.

			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.

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.

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

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]

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).

A340820 Decimal expansion of (1 + Product_{p prime} (1 - 2/(p*(p+1))))/2.

Original entry on oeis.org

7, 3, 5, 8, 4, 0, 3, 0, 6, 8, 0, 6, 4, 9, 8, 9, 3, 4, 0, 3, 7, 6, 1, 7, 8, 1, 6, 5, 4, 0, 2, 4, 1, 0, 4, 3, 7, 1, 2, 9, 6, 3, 1, 9, 1, 0, 0, 3, 4, 9, 3, 4, 4, 1, 8, 1, 7, 8, 6, 8, 6, 2, 7, 7, 0, 8, 8, 6, 6, 0, 5, 8, 3, 7, 9, 8, 4, 1, 3, 7, 2, 0, 4, 8, 1, 0, 5

Offset: 0

Amiram Eldar, Jan 22 2021

The asymptotic density of numbers k such that A046660(k) is even (A162644).

Detrey et al. (2016) calculated 1000 decimal digits of this constant.

			0.735840306806498934037617816540241043712963191003493...

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. See page 9.

Cf. A013661, A046660, A065472, A162644, A340818, A340819.

PARI
```
(prodeulerrat(1 - 2/(p*(p+1))) + 1)/2
```

Equals (1 + A065472/zeta(2))/2.

Equals lim_{n->oo} A340818(n)/A340819(n).

A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

Original entry on oeis.org

2, 3, 15, 35, 231, 5005, 255255, 1616615, 10140585, 462120945, 6685349671, 1236789689135, 30425026352721, 311494317420715, 13367169186706335, 1253429172199617105, 33151040519900217915, 3909612711980232366109, 119065478046670712967865, 7970583287524270870963077

Offset: 1

Amiram Eldar, Jul 26 2022

See A356093 for details.

Amiram Eldar, Table of n, a(n) for n = 1..351

Cf. A002110, A356093 (numerators).

Similar sequences: A038111, A338560, A340819, A341432, A342451, A342480.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Denominator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 20}]]

a(n) = denominator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import primorial, prime
def A356094(n): return (p:=primorial(n))//gcd(p,prime(n)-1) # Chai Wah Wu, Jul 26 2022

cp's OEIS Frontend

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

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A340820 Decimal expansion of (1 + Product_{p prime} (1 - 2/(p*(p+1))))/2.

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A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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