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%I #70 Apr 15 2023 15:43:01
%S 1,1,1,1,1,5,1,61,1,1385,5,50521,691,2702765,7,199360981,3617,
%T 19391512145,43867,2404879675441,174611,370371188237525,854513,
%U 69348874393137901,236364091,15514534163557086905,8553103,4087072509293123892361,23749461029,1252259641403629865468285
%N a(2n) = numerator of |Bernoulli(2n)|, a(2n+1) = Euler(2n).
%C Primes p which divide at least one a(n) for n<=p-2 are called weakly-irregular primes. For example, 19|a(11), 31|a(23), 37|a(32), 43|a(13), 47|a(15), 59|a(44), 61|a(7), ... - _Eric Chen_, Nov 26 2014
%C The weakly-irregular primes below 500 are 19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 359, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 467, 491. - _Eric Chen_, Nov 26 2014
%C A prime can divide more than one a(n) for n<=p-2; for example, 67 divides both a(27) and a(58); additional examples are 101, 149, 157, 241, 263, 307, 311, ... . - _Eric Chen_, Nov 26 2014
%C Smallest values of k such that the n-th weakly-irregular prime divides a(k) are 11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, ... . - _Eric Chen_, Nov 26 2014
%C Smallest prime factors (>= n+2) of a(n) are 1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 19, 691, 43, 1, 47, 3617, 228135437, 43867, 79, 283, 41737, 131, 31, 103, 2137, 657931, 67, 9349, 71, ... . - _Eric Chen_, Nov 26 2014
%C The irregular pairs are (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ... . - _Eric Chen_, Nov 26 2014
%H Eric Chen, <a href="/A246006/b246006.txt">Table of n, a(n) for n = 0..199</a>
%H Peter Luschny, <a href="http://www.luschny.de/math/primes/irregular.html"> Irregular primes</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Irregular_prime"> Irregular prime</a>
%e Euler(10) = 50521, so a(11) = 50521.
%e Bernoulli(12) = 691/2730, so a(12) = 691.
%t a246006[n_] := If[EvenQ[n], Abs[Numerator[BernoulliB[n]]], Abs[EulerE[n-1]]]; Table[a246006[n], {n, 0, 99}]
%o (Python)
%o from sympy import euler, bernoulli
%o def A246006(n): return abs(euler(n-1)) if n&1 else abs(bernoulli(n)).p # _Chai Wah Wu_, Apr 15 2023
%Y Cf. A027641, A122045, A000367, A028296, A000111, A000364, A000182.
%Y Cf. A050970, A050971, A068205.
%K nonn
%O 0,6
%A _Eric Chen_, Nov 13 2014
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