cp's OEIS Frontend

A321217 Genocchi irregular primes.

%I A321217 #28 Dec 06 2022 07:57:25
%S A321217 17,31,37,41,43,59,67,73,89,97,101,103,109,113,127,131,137,149,151,
%T A321217 157,193,223,229,233,241,251,257,263,271,277,281,283,293,307,311,313,
%U A321217 331,337,347,353,379,389,397,401,409,421,431,433,439,449,457,461,463,467,491,499
%N A321217 Genocchi irregular primes.
%C A321217 An odd prime p is G-irregular if it divides at least one of the integers G2, G4, ..., G(p-3).
%C A321217 Conjecture (Hu et al., 2019): The asymptotic density of this sequence within the primes is 1 - 3*A/(2*sqrt(e)) = 0.659776..., where A is Artin's constant (A005596). - _Amiram Eldar_, Dec 06 2022
%H A321217 Amiram Eldar, <a href="/A321217/b321217.txt">Table of n, a(n) for n = 1..1024</a>
%H A321217 Su Hu, Min-Soo Kim, Pieter Moree, and Min Sha, <a href="https://doi.org/10.1016/j.jnt.2019.03.012">Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture</a>, Journal of Number Theory, Vol. 205 (2019), pp. 59-80, <a href="https://arxiv.org/abs/1809.08431">preprint</a>, arXiv:1809.08431 [math.NT], 2019.
%H A321217 Peter Luschny, <a href="http://www.luschny.de/math/primes/irregular.html">Irregular Bernoulli and Euler Primes</a>.
%H A321217 Pieter Moree and Min Sha, <a href="https://doi.org/10.1017/S0004972719000443">Primes in arithmetic progressions and nonprimitive roots</a>, Bulletin of the Australian Mathematical Society (2019), <a href="https://arxiv.org/abs/1901.02650">preprint</a>, arXiv:1901.02650 [math.NT], 2019.
%H A321217 Pieter Moree and Pietro Sgobba, <a href="https://arxiv.org/abs/2209.08047">Prime divisors of l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l</a>, arXiv:2209.08047 [math.NT], 2022.
%p A321217 A321217_list := proc(bound)
%p A321217    local ae, F, p, m, maxp; F := NULL;
%p A321217    for m from 2 by 2 to bound do
%p A321217       p := nextprime(m+1);
%p A321217       ae := abs(m*euler(m-1, 0));
%p A321217       maxp := min(ae, bound);
%p A321217       while p <= maxp do
%p A321217           if ae mod p = 0 then F := F, p fi;
%p A321217           p := nextprime(p)
%p A321217       od
%p A321217    od;
%p A321217 sort({F}) end: A321217_list(500); # _Peter Luschny_, Nov 11 2018
%t A321217 G[n_] := G[n] = n EulerE[n - 1, 0];
%t A321217 GenocchiIrregularQ[p_] := AnyTrue[Table[G[k], {k, 2, p-3, 2}], Divisible[#, p]&];
%t A321217 Select[Prime[Range[2, 100]], GenocchiIrregularQ] (* _Jean-François Alcover_, Nov 16 2018 *)
%Y A321217 Cf. A036968 (Genocchi numbers), A000928 (irregular primes), A120337 (Euler-irregular primes), A128197 (strong irregular primes), A250216 (weak irregular primes), A005596.
%K A321217 nonn
%O A321217 1,1
%A A321217 _Michel Marcus_, Oct 31 2018
%E A321217 More terms from _Peter Luschny_, Nov 11 2018