A321217 Genocchi irregular primes.

17, 31, 37, 41, 43, 59, 67, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 149, 151, 157, 193, 223, 229, 233, 241, 251, 257, 263, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337, 347, 353, 379, 389, 397, 401, 409, 421, 431, 433, 439, 449, 457, 461, 463, 467, 491, 499

Offset

1,1

Comments

An odd prime p is G-irregular if it divides at least one of the integers G2, G4, ..., G(p-3).

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..1024

Su Hu, Min-Soo Kim, Pieter Moree, and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, Journal of Number Theory, Vol. 205 (2019), pp. 59-80, preprint, arXiv:1809.08431 [math.NT], 2019.

Peter Luschny, Irregular Bernoulli and Euler Primes.

Pieter Moree and Min Sha, Primes in arithmetic progressions and nonprimitive roots, Bulletin of the Australian Mathematical Society (2019), preprint, arXiv:1901.02650 [math.NT], 2019.

Pieter Moree and Pietro Sgobba, Prime divisors of l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l, arXiv:2209.08047 [math.NT], 2022.

Maple

A321217_list := proc(bound)
   local ae, F, p, m, maxp; F := NULL;
   for m from 2 by 2 to bound do
      p := nextprime(m+1);
      ae := abs(m*euler(m-1, 0));
      maxp := min(ae, bound);
      while p <= maxp do
          if ae mod p = 0 then F := F, p fi;
          p := nextprime(p)
      od
   od;
sort({F}) end: A321217_list(500); # Peter Luschny, Nov 11 2018

Mathematica

G[n_] := G[n] = n EulerE[n - 1, 0];
GenocchiIrregularQ[p_] := AnyTrue[Table[G[k], {k, 2, p-3, 2}], Divisible[#, p]&];
Select[Prime[Range[2, 100]], GenocchiIrregularQ] (* Jean-François Alcover, Nov 16 2018 *)

Crossrefs

Cf. A036968 (Genocchi numbers), A000928 (irregular primes), A120337 (Euler-irregular primes), A128197 (strong irregular primes), A250216 (weak irregular primes), A005596.

Sequence in context: A052006 A002675 A333855 * A095748 A350699 A235920

Adjacent sequences: A321214 A321215 A321216 * A321218 A321219 A321220

Keyword

nonn

Author

Michel Marcus, Oct 31 2018

Extensions

More terms from Peter Luschny, Nov 11 2018

Status

approved