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

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

Author

Amiram Eldar, Apr 13 2020

Keywords

• nonn
• frac

Examples

			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.
		

References

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

Links

• 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.

Crossrefs

Cf. A000984, A334074 (numerators).

Programs

Mathematica

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

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

Python

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

Formula

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