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%I #37 Apr 03 2023 10:36:11
%S 12,16,18,26,34,36,38,42,74,114,118,396,674,1870,4306,22808
%N Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) prime.
%C 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.
%C 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.
%C The number generated by n=4306 was recented proved prime. See Chris Caldwell's link for more details. - _T. D. Noe_, Apr 06 2009
%C a(17) > 5*10^4. - _Robert Price_, Oct 20 2013
%C a(17) > 74708. - _Simon Plouffe_, Mar 06 2022
%D S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.
%H Bernd Kellner, <a href="http://www.bernoulli.org/">Program Calcbn - A program for calculating Bernoulli numbers</a>
%H Chris Caldwell, <a href="https://t5k.org/top20/page.php?id=26">Top twenty irregular primes</a>
%H K. Ono, <a href="http://www.ams.org/notices/200606/fea-ono.pdf">Honoring a gift from Kumbakonam</a>, Notices Amer. Math. Soc., 53 (2006), 640-651.
%H Simon Plouffe, <a href="http://plouffe.fr/simon/articles/1607.0557v1.pdf">Primes as sums of irrational numbers</a>, Preprint, 2016.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrregularPrime.html">Irregular Prime</a>
%p 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
%t Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]
%Y Cf. A001067 (numerator of Bernoulli(2n)/(2n)).
%Y Cf. A033563 (primes in A001067).
%Y Cf. A092132 (n such that the numerator of Bernoulli(n) is prime).
%Y Cf. A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).
%Y Cf. A119766.
%K nonn,hard,more
%O 1,1
%A _T. D. Noe_, Sep 28 2005
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