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

1,3

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

Cf. A000984, A334074 (numerators).

Sequence in context: A087745 A087756 A243305 * A140351 A128444 A317745

Adjacent sequences: A334072 A334073 A334074 * A334076 A334077 A334078

Keyword

nonn,frac

Author

Amiram Eldar, Apr 13 2020

Status

approved