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

Offset: 1

Author

Peter Luschny, Jul 28 2019

Keywords

• sign
• tabf
• hard

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).

Examples

			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;
		

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

Crossrefs

Cf. A000367, A079294, A090947, A326726.

Programs

Sage

# See b-file.