cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A092133 Prime numerators of Bernoulli numbers, i.e., numerators of Bernoulli numbers with indices A092132.

Original entry on oeis.org

5, -691, 7, -3617, 43867, -26315271553053477373, 1520097643918070802691
Offset: 1

Views

Author

Eric W. Weisstein, Feb 22 2004

Keywords

Links

Crossrefs

Cf. A092132.

Programs

  • Mathematica
    Reap[For[n = 1, n <= 1000, n++, If[PrimeQ[nu = Numerator[BernoulliB[n]]], Print[nu]; Sow[nu]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2012 *)

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

Original entry on oeis.org

12, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808
Offset: 1

Views

Author

T. D. Noe, Sep 28 2005

Keywords

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

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[ # ]/# ]]&]

A250220 Numbers k such that A241601(k) is prime.

Original entry on oeis.org

7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, 337, 396, 455
Offset: 1

Views

Author

Eric Chen, Dec 24 2014

Keywords

Comments

Is the sequence infinite?
No other terms < 500. - Jinyuan Wang, Apr 02 2020

Crossrefs

Cf. A241601, A249909, A246006, A112548, A092132, A103234.

Extensions

a(17)-a(19) from Jinyuan Wang, Apr 02 2020

A132184 Numbers k such that the numerator of the Bernoulli number B(2k) ends with the digits 691.

Original entry on oeis.org

6, 21, 27, 321, 1266, 1527, 1821, 2526, 2576, 2721, 2950, 3126, 3246, 3426, 4206, 4236, 4821, 4926, 5286, 5721, 5946, 5950, 6100, 6351, 7018, 7138, 7172, 7386, 7806, 7931, 8037, 8790, 8796, 8826, 9021, 9048, 9426, 9478, 9726, 9921, 10221, 10326
Offset: 1

Views

Author

Alexander Adamchuk, Nov 04 2007

Keywords

Comments

The numerator of BernoulliB(12) is 691. The sequence gives semi-indices of the 691-automorphic numerators in the BernoulliB(n) sequence. All 4 initial terms are multiples of 3. Note that Bernoulli numerators corresponding to the first two terms are the automorphic primes: 691 and 1520097643918070802691.

Examples

			6 is a term because BernoulliB(2*6) = -691/2730.
21 is a term because BernoulliB(2*21) = 1520097643918070802691/1806.
27 is a term because BernoulliB(2*27) = 29149963634884862421418123812691/798.

Links

Crossrefs

Cf. A000367 (numerators of Bernoulli numbers B_2n).
Cf. A092132 (indices k of Bernoulli numbers B(k) whose numerators are primes).
Cf. A092133 (prime numerators of Bernoulli numbers).

Programs

  • Mathematica
    Do[ g=Numerator[ BernoulliB[ 2n ] ]; f=Mod[ Abs[ g ], 1000 ]; If[ f==691, Print[ n ] ], {n,1,1000}]
Select[Range[10400],Mod[Abs[Numerator[BernoulliB[2#]]],1000]==691&] (* Harvey P. Dale, May 05 2019 *)

Extensions

a(5)-a(42) from Donovan Johnson, Sep 05 2008

A250289 Numbers n such that the numerator of Bernoulli(n) (A027641(n)) is a semiprime.

Original entry on oeis.org

20, 24, 26, 34, 38, 40, 64, 72, 74, 114, 118, 144, 192
Offset: 1

Views

Author

Eric Chen, Dec 24 2014

Keywords

Examples

			a(1) = 20 so the numerator of Bernoulli(20) = 174611 = 283 * 617 is a semiprime.

Links

Crossrefs

Cf. A092132.
Showing 1-5 of 5 results.

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