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A334074
a(n) is the numerator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).
3
0, 0, 1, 1, 1, 1, 12, 1, 10, 71, 16, 103, 215, 311, 311, 311, 431, 30, 791, 36, 575, 8586, 222349, 222349, 182169, 144961, 747338, 8630, 1343, 89513, 2904968, 520321, 45746, 1005129, 350073, 1890784, 72480703, 34997904, 257894479, 257894479, 1755387611, 1755387611
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
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) = numerator(Sum_{p prime <= n, binomial(2*n, n) (mod p) > 0)} 1/p).
Lim_{k -> infinity} (1/k) Sum_{i=1..k} a(i)/A334075(i) = Sum_{k>=2} log(k)/2^k (A114124).
Lim_{k -> infinity} (1/k) Sum_{i=1..k} (a(i)/A334075(i))^2 = (Sum_{k>=2} log(k)/2^k)^2.
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
Cf. A000984, A114124, A334075 (denominators).
Sequence in context: A121985 A245839 A068329 * A010215 A059857 A322762
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Apr 13 2020
STATUS
approved