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

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

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

Table of n, a(n) for n=1..16.

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

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.

Sequence in context: A097620 A335222 A356863 * A032620 A224302 A096468

Adjacent sequences: A112545 A112546 A112547 * A112549 A112550 A112551

Keyword

nonn,hard,more

Author

T. D. Noe, Sep 28 2005

Status

approved