A340818 -id:A340818 - cp's OEIS frontend

A340819 Denominators of a sequence of fractions converging to A340820, the asymptotic density of numbers whose excess of prime divisors (A046660) is even (A162644).

6, 9, 54, 4, 264, 308, 7854, 374, 1564, 11339, 362848, 127541072, 1307295988, 2614591976, 42742894912, 1132686715168, 4608863185856, 71437379380768, 162734350229389504, 46216555465146619136, 427503138052606227008, 270181983249247135469056, 64347502466822129824768

Offset: 1

Amiram Eldar, Jan 22 2021

See A340818 for details.

Amiram Eldar, Table of n, a(n) for n = 1..462

Cf. A046660, A162644, A340818 (numerators), 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]; Denominator @ Array[f, 30]

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).

A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 8, 3, 1, 2, 1, 6, 4, 1, 1, 2, 1, 2, 1, 1, 12, 1, 1, 4, 16, 10, 1, 1, 18, 8, 3, 1, 4, 1, 2, 5, 2, 27, 1, 2, 1, 6, 1, 32, 14, 3, 1, 1, 1, 2, 4, 1, 8, 25, 128, 1, 2, 9, 2, 4, 1, 2, 3, 1, 4, 2, 1, 8, 1, 2, 16, 1, 1, 2, 9, 1, 2, 6, 40, 4, 1, 2, 1

Offset: 1

Amiram Eldar, Jul 26 2022

f(n) = a(n)/A356094(n) is the asymptotic density of numbers k such that prime(n) = A053669(k) is the smallest prime not dividing k.

The asymptotic mean of A053669 is 2.9200509773... (A249270) which is the weighted mean of the primes with f(n) as weights. The corresponding asymptotic standard deviation, which can be evaluated from the second raw moment Sum_{n>=1} f(n) * prime(n)^2, is 2.8013781465... .

			Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).

Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]

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

from math import gcd
from sympy import prime, primorial
def A356093(n): return (p:=prime(n)-1)//gcd(p,primorial(n)) # Chai Wah Wu, Jul 26 2022

a(n) = 1 iff prime(n) is in A039787.
Let f(n) = a(n)/A356094(n):
f(n) = A006093(n)/A002110(n).
Sum_{n>=1} f(n) = 1.
Sum_{n>=1} f(n) * prime(n) = A249270.

cp's OEIS Frontend

A340819 Denominators 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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A356093 a(n) = numerator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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