%I A112548
%S A112548 12,16,18,26,34,36,38,42,74,114,118,396,674,1870,4306,22808
%N A112548 Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) 
               prime.
%C A112548 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 A112548 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 A112548 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]
%D A112548 K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 
               (2006), 640-651.
%D A112548 S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. 
               Soc., 3 (1911), 219-234.
%H A112548 Bernd Kellner, <a href="http://www.bernoulli.org/">Program Calcbn - A 
               program for calculating Bernoulli numbers</a>
%H A112548 Chris Caldwell, <a href="http://primes.utm.edu/top20/page.php?id=26">
               Top twenty irregular primes</a>
%H A112548 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IrregularPrime.html">Irregular Prime</a>
%p A112548 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
%t A112548 Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]
%Y A112548 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).
%Y A112548 Cf. A119766.
%Y A112548 Sequence in context: A043544 A097620 A089021 this_sequence A032620 A096468 
               A054281
%Y A112548 Adjacent sequences: A112545 A112546 A112547 this_sequence A112549 A112550 
               A112551
%K A112548 hard,nonn
%O A112548 1,1
%A A112548 T. D. Noe (noe(AT)sspectra.com), Sep 28 2005