OFFSET
1,3
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(2n, 𝑛), Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 83-92.
FORMULA
a(n) = denominator(Sum_{p prime <= n, binomial(2*n, n) (mod p) > 0)} 1/p).
EXAMPLE
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.
MATHEMATICA
a[n_] := Denominator[Plus @@ (1/Select[Range[n], PrimeQ[#] && !Divisible[Binomial[2n, n], #] &])]; Array[a, 50]
PROG
(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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Apr 13 2020
STATUS
approved