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A112548 Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) prime. 8
12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808 (list; graph; refs; listen; history; text; internal format)



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]

a(17) > 5*10^4. - Robert Price, Oct 20 2013


Simon Plouffe, Primes as sums of irrational numbers, Preprint. 2016; http://plouffe.fr/simon/articles/1607.0557v1.pdf

S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.


Table of n, a(n) for n=1..16.

Bernd Kellner, Program Calcbn - A program for calculating Bernoulli numbers

Chris Caldwell, Top twenty irregular primes

K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 (2006), 640-651.

Eric Weisstein's World of Mathematics, Irregular Prime


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


Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]


Cf. A001067 (numerator of Bernoulli(2n)/(2n)).

Cf. A033563 (primes in A001067).

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).

Cf. A119766.

Sequence in context: A051518 A043544 A097620 * A032620 A224302 A096468

Adjacent sequences:  A112545 A112546 A112547 * A112549 A112550 A112551




T. D. Noe, Sep 28 2005



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Last modified October 15 13:38 EDT 2019. Contains 328030 sequences. (Running on oeis4.)