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

Author

Stefan Krämer, Jun 22 2006

Keywords

• nonn

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

Examples

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

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.
• Romeo 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.

Crossrefs

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

Programs

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 *)

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).

Extensions

Terms 251 through 983 from Peter Luschny, Apr 25 2011