OFFSET
1,1
COMMENTS
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
a(17) > 74708. - Simon Plouffe, Mar 06 2022
REFERENCES
S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.
LINKS
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.
Simon Plouffe, Primes as sums of irrational numbers, Preprint, 2016.
Eric Weisstein's World of Mathematics, Irregular Prime
MAPLE
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
MATHEMATICA
Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
T. D. Noe, Sep 28 2005
STATUS
approved