A326726 The prime factorization of abs(E(2k)) for k >= 2, E(k) the k-th Euler number. Factors sorted by size with the smallest factor negated. a(n) = -1 by convention for n = 1, 2.

-1, -1, -5, -61, -5, 277, -19, 2659, -5, 13, 43, 967, -47, 4241723, -5, 17, 228135437, -79, 349, 87224971, -5, 5, 41737, 354957173, -31, 1567103, 1427513357, -5, 13, 2137, 111691689741601, -67, 61001082228255580483, -5, 19, 29, 71, 30211, 2717447, 77980901

Offset: 1

Author

Peter Luschny, Jul 29 2019

Keywords

• sign
• tabf
• hard

Comments

For small Euler numbers the factorizations were computed with SageMath, see the b-file for the script. For larger Euler 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 E(164) (the prime factors of E(166) 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], [-5], [-61], [-5, 277], [-19, 2659], [-5, 13, 43, 967], [-47, 4241723], [-5, 17, 228135437], [-79, 349, 87224971], [-5, 5, 41737, 354957173], ... ].
The first few factorizations are:
E(4)  = 5;
E(6)  = 61;
E(8)  = 5 * 277;
E(10) = 19 * 2659;
E(12) = 5 * 13 * 43 * 967;
E(14) = 47 * 4241723;
E(16) = 5 * 17 * 228135437;
E(18) = 79 * 349 * 87224971;
E(20) = 5 * 5 * 41737 * 354957173;
E(22) = 31 * 1567103 * 1427513357;
E(24) = 5 * 13 * 2137 * 111691689741601;
		

Links

• Peter Luschny, Table of n, a(n) for n = 1..428
• factordb, Status of E(166).
• S. S. Wagstaff, Prime factors of the absolute values of Euler numbers

Crossrefs

Cf. A122045, A000364, A326727.

Programs

Sage

# See b-file.