A334074 -id:A334074 - cp's OEIS frontend

A114124 Decimal expansion of logarithm of A112302.

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

Offset: 0

Eric W. Weisstein, Feb 08 2006

			0.5078339228684383921890418407220763742462184334326009...

Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 4.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, Volume 155, October 2015, Pages 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics (2019) Vol. 74, No. 1, 6.
Paul Erdős, Ronald L. Graham, Imre Z. Ruzsa and Ernst G. Straus, On the prime factors of C(2n, n), Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 83-92.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 183.
Yusuke Kobayashi and Ryoga Mahara, Approximation algorithm for Steiner tree problem with neighbor-induced cost, J. Operations Res. Soc. Japan, (2023) Vol. 66, No. 1, 18-36. See p. 32.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, Volume 436, Issue 1, 1 April 2016, Pages 513-520.

Cf. A000984, A014963, A112302, A334074, A334075.

First@ RealDigits[-Derivative[1, 0][PolyLog][0, 1/2], 10, 105] (* Eric W. Weisstein, edited by Michael De Vlieger, Jan 21 2019 *)

suminf(n=2,log(n)>>n) \\ Charles R Greathouse IV, Sep 08 2014

Equals Sum_{n>=2} log(n)/2^n. - Jean-François Alcover, Apr 14 2014
Equals lim_{k->oo} (1/k) Sum_{i=1..k} A334074(i)/A334075(i). - Amiram Eldar, Apr 14 2020
Equals Sum_{n>=1} Lambda(n)/(2^n-1), where Lambda(n) = log(A014963(n)) is the Mangoldt function. - Amiram Eldar, Jul 07 2021
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} f(k), where f(k) = Sum_{primes p <= k, binomial(2*k,k) mod p != 0} 1/p = A334074(k)/A334075(k) (Erdős et al., 1975). - Amiram Eldar, May 25 2025

A334075 a(n) is the denominator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 35, 7, 21, 105, 55, 165, 429, 1001, 1001, 1001, 1547, 221, 4199, 323, 2261, 24871, 572033, 572033, 408595, 312455, 937365, 17043, 8671, 130065, 4032015, 1344005, 227447, 3866599, 840565, 2521695, 93302715, 118183439, 419014011, 419014011, 5726524817

Offset: 1

Amiram Eldar, Apr 13 2020

			For n = 7, binomial(2*7, 7) = 3432 = 2^3 * 3 * 11 * 13, and there are 2 primes p <= 7 which are not divisors of 3432: 5 and 7. Therefore, a(7) = denominator(1/5 + 1/7) = denominator(12/35) = 35.

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.

Chai Wah Wu, Table of n, a(n) for n = 1..3844
Paul Erdős, Ronald L. Graham, Imre Z. Ruzsa and Ernst G. Straus, On the prime factors of C(2*n, n), Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 83-92.

Cf. A000984, A334074 (numerators).

a[n_] := Denominator[Plus @@ (1/Select[Range[n],PrimeQ[#] && !Divisible[Binomial[2n, n],#] &])]; Array[a, 50]

a(n) = {my(s=0, b=binomial(2*n,n)); forprime(p=2, n, if (b % p, s += 1/p)); denominator(s);} \\ Michel Marcus, Apr 14 2020

from fractions import Fraction
from sympy import binomial, isprime
def A334075(n):
    b = binomial(2*n,n)
    return sum(Fraction(1,p) for p in range(2,n+1) if b % p != 0 and isprime(p)).denominator # Chai Wah Wu, Apr 14 2020

a(n) = denominator(Sum_{p prime <= n, binomial(2*n, n) (mod p) > 0} 1/p).

cp's OEIS Frontend

A114124 Decimal expansion of logarithm of A112302.

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