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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.

%I A326727 #26 Aug 20 2019 15:42:15
%S A326727 -1,-1,-1,-1,-1,-5,-691,-7,-3617,-43867,-283,617,-11,131,593,-103,
%T A326727 2294797,-13,657931,-7,9349,362903,-5,1721,1001259881,-37,683,
%U A326727 305065927,-17,151628697551,-26315271553053477373,-19,154210205991661,-137616929,1897170067619
%N 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.
%C A326727 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..
%C A326727 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 A326727 The factorizations listed in the b-file currently go up to B(204) (the prime factors of numerator(B(206)) are not yet known).
%H A326727 Peter Luschny, <a href="/A326727/b326727.txt">Table of n, a(n) for n = 1..460</a>
%H A326727 factordb, <a href="http://factordb.com/index.php?showid=1100000000705090099">Status of numerator(B(206))</a>.
%H A326727 S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/bnum">Prime factors of the absolute values of Bernoulli numerators</a>
%e A326727 The data is given as a flatted list of factorizations written with the conventions
%e A326727 stated above. Because it is a list the offset is 1. The list starts:
%e A326727 [[-1], [-1], [-1], [-1], [-1], [-5], [-691], [-7], [-3617], [-43867], [-283, 617], [-11, 131, 593], [-103, 2294797], [-13, 657931], [-7, 9349, 362903], ... ].
%e A326727 .
%e A326727 The first few factorizations are:
%e A326727 B(10) = 5;
%e A326727 B(12) = 691;
%e A326727 B(14) = 7;
%e A326727 B(16) = 3617;
%e A326727 B(18) = 43867;
%e A326727 B(20) = 283 * 617;
%e A326727 B(22) = 11 * 131 * 593;
%e A326727 B(24) = 103 * 2294797;
%e A326727 B(26) = 13 * 657931;
%e A326727 B(28) = 7 * 9349 * 362903;
%e A326727 B(30) = 5 * 1721 * 1001259881;
%o A326727 (Sage) # See b-file.
%Y A326727 Cf. A000367, A079294, A090947, A326726.
%K A326727 sign,tabf,hard
%O A326727 1,6
%A A326727 _Peter Luschny_, Jul 28 2019