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A334074 a(n) is the numerator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).

%I A334074 #22 Apr 25 2025 04:29:48
%S A334074 0,0,1,1,1,1,12,1,10,71,16,103,215,311,311,311,431,30,791,36,575,8586,
%T A334074 222349,222349,182169,144961,747338,8630,1343,89513,2904968,520321,
%U A334074 45746,1005129,350073,1890784,72480703,34997904,257894479,257894479,1755387611,1755387611
%N A334074 a(n) is the numerator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).
%C A334074 Erdős et al. (1975) could not decide if the fraction f(n) = a(n)/A334075(n) is bounded. They found its asymptotic mean (see formula).
%D A334074 R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.
%H A334074 Chai Wah Wu, <a href="/A334074/b334074.txt">Table of n, a(n) for n = 1..3844</a>
%H A334074 Paul Erdős, Ronald L. Graham, Imre Z. Ruzsa and Ernst G. Straus, <a href="https://doi.org/10.1090/S0025-5718-1975-0369288-3">On the prime factors of C(2*n, n)</a>, Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 83-92.
%F A334074 a(n) = numerator(Sum_{p prime <= n, binomial(2*n, n) (mod p) > 0} 1/p).
%F A334074 Limit_{k -> infinity} (1/k) Sum_{i=1..k} a(i)/A334075(i) = Sum_{k>=2} log(k)/2^k (A114124).
%F A334074 Limit_{k -> infinity} (1/k) Sum_{i=1..k} (a(i)/A334075(i))^2 = (Sum_{k>=2} log(k)/2^k)^2.
%e A334074 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) = numerator(1/5 + 1/7) = numerator(12/35) = 12.
%t A334074 a[n_] := Numerator[Plus @@ (1/Select[Range[n],PrimeQ[#] && !Divisible[Binomial[2n, n],#] &])]; Array[a, 50]
%o A334074 (PARI) a(n) = {my(s=0, b=binomial(2*n,n)); forprime(p=2, n, if (b % p, s += 1/p)); numerator(s);} \\ _Michel Marcus_, Apr 14 2020
%o A334074 (Python)
%o A334074 from fractions import Fraction
%o A334074 from sympy import binomial, isprime
%o A334074 def A334074(n):
%o A334074     b = binomial(2*n,n)
%o A334074     return sum(Fraction(1,p) for p in range(2,n+1) if b % p != 0 and isprime(p)).numerator # _Chai Wah Wu_, Apr 14 2020
%Y A334074 Cf. A000984, A114124, A334075 (denominators).
%K A334074 nonn,frac
%O A334074 1,7
%A A334074 _Amiram Eldar_, Apr 13 2020