login
A334075
a(n) is the denominator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).
3
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
1,3
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.
LINKS
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
Cf. A000984, A334074 (numerators).
Sequence in context: A087745 A087756 A243305 * A140351 A128444 A317745
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Apr 13 2020
STATUS
approved