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A250216 Weak irregular primes. A prime is weak irregular iff it is a Bernoulli irregular prime or an Euler irregular prime.

%I A250216 #15 Apr 03 2023 10:36:13
%S A250216 19,31,37,43,47,59,61,67,71,79,101,103,131,137,139,149,157,193,223,
%T A250216 233,241,251,257,263,271,277,283,293,307,311,347,349,353,359,373,379,
%U A250216 389,401,409,419,421,433,461,463,467,491,509,523,541,547,557,563,571,577,587,593
%N A250216 Weak irregular primes. A prime is weak irregular iff it is a Bernoulli irregular prime or an Euler irregular prime.
%C A250216 Primes p which divide A241601(k) for some k.
%H A250216 Peter Luschny, <a href="http://www.luschny.de/math/primes/irregular.html">Irregular Bernoulli and Euler Primes</a>.
%H A250216 The Prime Pages, <a href="https://t5k.org/top20/page.php?id=26">Irregular Primes</a>
%H A250216 The Prime Pages, <a href="https://t5k.org/top20/page.php?id=25">Euler Irregular</a>
%t A250216 pmax = 593; m0 = 200; dm = 100;
%t A250216 b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
%t A250216 c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
%t A250216 (* a1 = A241601 *) a1[0] = 1; a1[n_] := a1[n] = If[EvenQ[n], b[n/2] // Abs, c[(n - 1)/2]];
%t A250216 f[m_] := f[m] = Module[{}, aa = Table[a1[n], {n, 0, m}]; okQ[p_] := AnyTrue[aa, Divisible[#, p] &]; Reap[For[p = 2, p <= pmax, p = NextPrime[p], If[okQ[p], Sow[p]]]][[2, 1]]];
%t A250216 f[m = m0]; f[m = m + dm];
%t A250216 While[Print["m = ", m]; f[m] != f[m - dm], m = m + dm];
%t A250216 A250216 = f[m] (* _Jean-François Alcover_, Jul 23 2019 *)
%Y A250216 Cf. A000928, A120337, A128197.
%K A250216 nonn
%O A250216 1,1
%A A250216 _Eric Chen_, Dec 24 2014