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A112548
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Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) prime.
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8
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12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808
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OFFSET
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1,1
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COMMENTS
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In 1911 Ramanujan believed that the numerator of Bernoulli(n)/n for n even was (apart from sign) always either 1 or a prime. This is false.
Equivalently, n such that the numerator of zeta(1-n) is prime. No other n < 23000. Kellner's Calcbn program was used to generate the numerators of Bernoulli(k)/k for k > 5000. Mathematica and PFGW were used to test for probable primes. David Broadhurst found n=4306, which yields a 10342-digit probable prime. For n<4306, the primes have been proved. Bouk de Water proved the prime for n=1870. All these primes are necessarily irregular.
The number generated by n=4306 was recented proved prime. See Chris Caldwell's link for more details. - T. D. Noe, Apr 06 2009
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REFERENCES
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S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.
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LINKS
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MAPLE
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A112548 := proc(nmax) local numr; for n from 2 to nmax by 2 do numr := abs(numer(bernoulli(n)/n)) ; if isprime(numr) then print(n) ; fi ; od ; end : A112548(3000) ; # R. J. Mathar, Jun 21 2006
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MATHEMATICA
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Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]
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CROSSREFS
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Cf. A001067 (numerator of Bernoulli(2n)/(2n)).
Cf. A092132 (n such that the numerator of Bernoulli(n) is prime).
Cf. A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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