The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A112548 Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) prime. 8
12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808 (list; graph; refs; listen; history; text; internal format)



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


S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.


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


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


Select[Range[2, 5000, 2], PrimeQ[Numerator[BernoulliB[ # ]/# ]]&]


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: A043544 A097620 A335222 * A032620 A224302 A096468

Adjacent sequences:  A112545 A112546 A112547 * A112549 A112550 A112551




T. D. Noe, Sep 28 2005



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)