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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.

%I A326726 #14 Aug 20 2019 15:54:34
%S A326726 -1,-1,-5,-61,-5,277,-19,2659,-5,13,43,967,-47,4241723,-5,17,
%T A326726 228135437,-79,349,87224971,-5,5,41737,354957173,-31,1567103,
%U A326726 1427513357,-5,13,2137,111691689741601,-67,61001082228255580483,-5,19,29,71,30211,2717447,77980901
%N 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.
%C A326726 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..
%C A326726 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.)
%C A326726 The factorizations listed in the b-file currently go up to E(164) (the prime factors of E(166) are not yet known).
%H A326726 Peter Luschny, <a href="/A326726/b326726.txt">Table of n, a(n) for n = 1..428</a>
%H A326726 factordb, <a href="http://factordb.com/index.php?showid=1100000000704918270">Status of E(166)</a>.
%H A326726 S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/enum">Prime factors of the absolute values of Euler numbers</a>
%e A326726 The data is given as a flatted list of factorizations written with the conventions
%e A326726 stated above. Because it is a list the offset is 1. The list starts:
%e A326726 [[-1], [-1], [-5], [-61], [-5, 277], [-19, 2659], [-5, 13, 43, 967], [-47, 4241723], [-5, 17, 228135437], [-79, 349, 87224971], [-5, 5, 41737, 354957173], ... ].
%e A326726 The first few factorizations are:
%e A326726 E(4)  = 5;
%e A326726 E(6)  = 61;
%e A326726 E(8)  = 5 * 277;
%e A326726 E(10) = 19 * 2659;
%e A326726 E(12) = 5 * 13 * 43 * 967;
%e A326726 E(14) = 47 * 4241723;
%e A326726 E(16) = 5 * 17 * 228135437;
%e A326726 E(18) = 79 * 349 * 87224971;
%e A326726 E(20) = 5 * 5 * 41737 * 354957173;
%e A326726 E(22) = 31 * 1567103 * 1427513357;
%e A326726 E(24) = 5 * 13 * 2137 * 111691689741601;
%o A326726 (Sage) # See b-file.
%Y A326726 Cf. A122045, A000364, A326727.
%K A326726 sign,tabf,hard
%O A326726 1,3
%A A326726 _Peter Luschny_, Jul 29 2019