OFFSET
1,7
COMMENTS
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).
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
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) = numerator(1/5 + 1/7) = numerator(12/35) = 12.
MATHEMATICA
a[n_] := Numerator[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)); numerator(s); } \\ Michel Marcus, Apr 14 2020
(Python)
from fractions import Fraction
from sympy import binomial, isprime
def A334074(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)).numerator # Chai Wah Wu, Apr 14 2020
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Apr 13 2020
STATUS
approved