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
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 31 2018
EXTENSIONS
More terms from Peter Luschny, Nov 11 2018
STATUS
approved