cp's OEIS Frontend

A112548 Numbers k such that the numerator of Bernoulli(k)/k is (apart from sign) prime.

%I A112548 #47 Jun 28 2025 18:01:37
%S A112548 12,16,18,26,34,36,38,42,74,114,118,396,674,1870,4306,22808
%N A112548 Numbers k such that the numerator of Bernoulli(k)/k is (apart from sign) prime.
%C A112548 In 1911 Ramanujan believed that the numerator of Bernoulli(k)/k for k even was (apart from sign) always either 1 or a prime. This is false.
%C A112548 Equivalently, k such that the numerator of zeta(1-k) is prime. No other k < 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 k=4306 was recently proved prime. See Chris Caldwell's link for more details. - _T. D. Noe_, Apr 06 2009
%C A112548 a(17) > 50000. - _Robert Price_, Oct 20 2013
%C A112548 a(17) > 74708. - _Simon Plouffe_, Mar 06 2022
%C A112548 a(17) > 270000. - _Serge Batalov_, Jun 26 2025
%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="https://t5k.org/top20/page.php?id=26">Top twenty irregular primes</a>
%H A112548 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 A112548 Simon Plouffe, <a href="http://plouffe.fr/simon/articles/1607.0557v1.pdf">Primes as sums of irrational numbers</a>, Preprint, 2016.
%H A112548 Eric Weisstein's World of Mathematics, <a href="https://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_, Jun 21 2006
%t A112548 Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]
%Y A112548 Cf. A001067 (numerator of Bernoulli(2n)/(2n)).
%Y A112548 Cf. A033563 (primes in A001067).
%Y A112548 Cf. A092132 (n such that the numerator of Bernoulli(n) is prime).
%Y A112548 Cf. A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).
%Y A112548 Cf. A119766.
%K A112548 nonn,hard,more
%O A112548 1,1
%A A112548 _T. D. Noe_, Sep 28 2005