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

12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808

Offset: 1

Author

T. D. Noe, Sep 28 2005

Keywords

• nonn
• hard
• more

Comments

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.

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.

The number generated by k=4306 was recently proved prime. See Chris Caldwell's link for more details. - T. D. Noe, Apr 06 2009

a(17) > 50000. - Robert Price, Oct 20 2013

a(17) > 74708. - Simon Plouffe, Mar 06 2022

a(17) > 270000. - Serge Batalov, Jun 26 2025

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

Crossrefs

Cf. A001067 (numerator of Bernoulli(2n)/(2n)).

Cf. A033563 (primes in A001067).

Cf. A092132 (n such that the numerator of Bernoulli(n) is prime).

Cf. A112741 (primes p such that zeta(1-2p)/zeta(-1) is prime).

Cf. A119766.

Programs

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[ # ]/# ]]&]