A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, 619, 677, 691, 709, 739, 751, 761, 769, 773, 811, 821, 877, 887, 907, 929, 941, 967, 971, 983

Offset

1,1

Comments

Conjecture (Ernvall and Metsänkylä, 1978): The asymptotic density of this sequence within the primes is 1 - 1/sqrt(e) = 0.393469... (A290506), the same as the corresponding conjectured density of the irregular primes (A000928). - Amiram Eldar, Dec 06 2022

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Reijo Ernvall, On the distribution mod 8 of the E-irregular primes, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, Vol. 1, 1975, pp. 195-198.

Reijo Ernvall and Tauno Metsänkylä, Cyclotomic invariants and 𝐸-irregular primes, Mathematics of Computation, Vol. 32, No. 142 (1978), pp. 617-629; Corrigenda, ibid., Vol. 33, No. 145 (1979), p. 433.

Su Hu and Min-Soo Kim, A note on the irregular primes with respect to Euler polynomials, arXiv:1510.01558 [math.NT], 2015.

Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018.

R. Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From N. J. A. Sloane, Jan 25 2013

Prime Pages, Euler Irregular

Samuel S. Wagstaff, Prime divisors of the Bernoulli and Euler numbers, Number theory for the millennium, III, 2002, pp. 357-374, 2002. MR 1956285.

Formula

The (trivial) divisors of E(2n) are given by the theorem of Sylvester (1861): Let p prime with p=1 (mod 4), p-1|2n, p^k|2n then p^{k+1} | E(2n).

Example

a(1) = 19 because 19 divides E(10) = -19*2659 and 10 + 1 < 19.

Maple

A120337_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(euler(m));
  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: # Peter Luschny, Apr 25 2011

Mathematica

fQ[p_] := Block[{k = 1}, While[ 2k +1 < p && Mod[ EulerE[ 2k], p] != 0, k++]; p > 2k +1]; Select[ Prime@ Range@ 168, fQ@# &] (* Robert G. Wilson v, Dec 10 2014 *)

Crossrefs

Cf. A000928, A092218, A120115, A122045, A290506.

Sequence in context: A335418 A164320 A154418 * A120115 A157995 A043298

Adjacent sequences: A120334 A120335 A120336 * A120338 A120339 A120340

Keyword

nonn

Author

Stefan Krämer, Jun 22 2006

Extensions

Terms 251 through 983 from Peter Luschny, Apr 25 2011

Status

approved