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A334075
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a(n) is the denominator of the sum of reciprocals of primes not exceeding n and not dividing binomial(2*n, n).
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3
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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
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OFFSET
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1,3
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LINKS
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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.
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FORMULA
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a(n) = denominator(Sum_{p prime <= n, binomial(2*n, n) (mod p) > 0)} 1/p).
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EXAMPLE
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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.
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MATHEMATICA
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a[n_] := Denominator[Plus @@ (1/Select[Range[n], PrimeQ[#] && !Divisible[Binomial[2n, n], #] &])]; Array[a, 50]
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PROG
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(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
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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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