1,1

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. [From T. D. Noe (noe(AT)sspectra.com), Apr 06 2009]

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

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

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

Chris Caldwell, Top twenty irregular primes

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 (mathar(AT)strw.leidenuniv.nl), Jun 21 2006

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

Cf. A001067 (numerator of Bernoulli(2n)/(2n)), A033563 (primes in A001067), A092132 (n such that the numerator of Bernoulli(n) is prime), A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).

Cf. A119766.

Sequence in context: A043544 A097620 A089021 this_sequence A032620 A096468 A054281

Adjacent sequences: A112545 A112546 A112547 this_sequence A112549 A112550 A112551

hard,nonn

T. D. Noe (noe(AT)sspectra.com), Sep 28 2005