A326727 - OEIS

A326727

The prime factorization of abs(numerator(B(2k))) for k >= 5, B(k) the k-th Bernoulli number. Factors sorted by size with the smallest factor negated. a(n) = -1 by convention for 1 <= n <= 5.

-1, -1, -1, -1, -1, -5, -691, -7, -3617, -43867, -283, 617, -11, 131, 593, -103, 2294797, -13, 657931, -7, 9349, 362903, -5, 1721, 1001259881, -37, 683, 305065927, -17, 151628697551, -26315271553053477373, -19, 154210205991661, -137616929, 1897170067619

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

For small Bernoulli numbers the factorizations were computed with SageMath, see the b-file for the script. For larger Bernoulli numbers the values were taken from the table of S. S. Wagstaff, Jr..

The smallest factor was negated only to be able to distinguish the individual factorizations easily. (No general formula for the number of factors is known.)

The factorizations listed in the b-file currently go up to B(204) (the prime factors of numerator(B(206)) are not yet known).

LINKS

Peter Luschny, Table of n, a(n) for n = 1..460

factordb, Status of numerator(B(206)).

S. S. Wagstaff, Prime factors of the absolute values of Bernoulli numerators

EXAMPLE

The data is given as a flatted list of factorizations written with the conventions

stated above. Because it is a list the offset is 1. The list starts:

[[-1], [-1], [-1], [-1], [-1], [-5], [-691], [-7], [-3617], [-43867], [-283, 617], [-11, 131, 593], [-103, 2294797], [-13, 657931], [-7, 9349, 362903], ... ].

The first few factorizations are:

B(10) = 5;

B(12) = 691;

B(14) = 7;

B(16) = 3617;

B(18) = 43867;

B(20) = 283 * 617;

B(22) = 11 * 131 * 593;

B(24) = 103 * 2294797;

B(26) = 13 * 657931;

B(28) = 7 * 9349 * 362903;

B(30) = 5 * 1721 * 1001259881;