A128197 -id:A128197 - cp's OEIS frontend

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

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, 509, 523, 541, 547, 557, 563, 571, 577, 587, 593

Offset: 1

Eric Chen, Dec 24 2014

nonn

Primes p which divide A241601(k) for some k.

Peter Luschny, Irregular Bernoulli and Euler Primes.
The Prime Pages, Irregular Primes
The Prime Pages, Euler Irregular

Cf. A000928, A120337, A128197.

pmax = 593; m0 = 200; dm = 100;
b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
(* a1 = A241601 *) a1[0] = 1; a1[n_] := a1[n] = If[EvenQ[n], b[n/2] // Abs, c[(n - 1)/2]];
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]]];
f[m = m0]; f[m = m + dm];
While[Print["m = ", m]; f[m] != f[m - dm], m = m + dm];
A250216 = f[m] (* Jean-François Alcover, Jul 23 2019 *)

A321217 Genocchi irregular primes.

Original entry on oeis.org

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

Michel Marcus, Oct 31 2018

nonn

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

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.

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

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

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 *)

More terms from Peter Luschny, Nov 11 2018

A250214 Number of values of k such that prime(n) divides A241601(k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 1, 2, 0, 1, 1, 0, 1, 1, 3, 3, 0, 0, 0, 0, 1, 2, 3, 1, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 4, 0, 0, 1, 0, 1, 2, 2, 1, 2, 0, 1, 3, 3, 1, 0, 0, 1, 1, 3, 3, 2, 0, 0, 3, 1, 1

Offset: 1

Eric Chen, Dec 26 2014

nonn

a(n) is called the weak irregular index of n-th prime, that is, the Bernoulli irregular index + Euler irregular index.
Prime(n) is a regular prime if and only if a(n) = 0.
Does every natural number appear in this sequence? For example, for the primes 491 and 1151, a(94) = a(190) = 4. (491 and 1151 are the only primes below 1800 with weak irregular index 4 or more.) However, does a(n) have a limit?

			a(8) = 1 since the 8th prime is 19, which divides A241601(11).
a(13) = 0 since the 13th prime is 41, a regular prime.
a(19) = 2 since the 19th prime is 67, which divides both A241601(27) and A241601(58).

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A091888, A091887, A250216, A000928, A120337, A128197, A250213.

A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

Original entry on oeis.org

11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, 133, 84, 211, 127, 164, 100, 84, 9, 20, 156, 88, 87, 280, 19, 71, 125, 163, 100, 200, 382, 126, 159, 240, 215, 196, 130, 94, 292, 141, 400, 86, 270, 222, 175, 389, 52, 45, 22, 592, 522, 20

Offset: 1

Eric Chen, Dec 26 2014

nonn

A prime p can divide A241601(k) for more than one k; the first few examples are as follows:
p      k
   27, 58
  63, 68
  130, 147
  62, 110
  211, 239
  100, 213
  88, 91, 137
  87, 193, 292
  19, 257
  71, 186, 300
etc.

			19 is the first weak irregular prime and divides A241601(11) = 50521, so a(1) = 11.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A035112, A120337, A128197, A250216.

More terms from Jinyuan Wang, Feb 18 2022

cp's OEIS Frontend

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

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A321217 Genocchi irregular primes.

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A250214 Number of values of k such that prime(n) divides A241601(k).

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A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

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