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

%I A334075 #22 Apr 25 2025 04:29:51
%S A334075 1,1,3,3,5,5,35,7,21,105,55,165,429,1001,1001,1001,1547,221,4199,323,
%T A334075 2261,24871,572033,572033,408595,312455,937365,17043,8671,130065,
%U A334075 4032015,1344005,227447,3866599,840565,2521695,93302715,118183439,419014011,419014011,5726524817
%N A334075 a(n) is the denominator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).
%D A334075 R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.
%H A334075 Chai Wah Wu, <a href="/A334075/b334075.txt">Table of n, a(n) for n = 1..3844</a>
%H A334075 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 A334075 a(n) = denominator(Sum_{p prime <= n, binomial(2*n, n) (mod p) > 0} 1/p).
%e A334075 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.
%t A334075 a[n_] := Denominator[Plus @@ (1/Select[Range[n],PrimeQ[#] && !Divisible[Binomial[2n, n],#] &])]; Array[a, 50]
%o A334075 (PARI) 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
%o A334075 (Python)
%o A334075 from fractions import Fraction
%o A334075 from sympy import binomial, isprime
%o A334075 def A334075(n):
%o A334075     b = binomial(2*n,n)
%o A334075     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
%Y A334075 Cf. A000984, A334074 (numerators).
%K A334075 nonn,frac
%O A334075 1,3
%A A334075 _Amiram Eldar_, Apr 13 2020