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

%I A120337 #46 Apr 25 2025 04:29:39
%S A120337 19,31,43,47,61,67,71,79,101,137,139,149,193,223,241,251,263,277,307,
%T A120337 311,349,353,359,373,379,419,433,461,463,491,509,541,563,571,577,587,
%U A120337 619,677,691,709,739,751,761,769,773,811,821,877,887,907,929,941,967,971,983
%N A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.
%C A120337 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
%H A120337 Robert G. Wilson v, <a href="/A120337/b120337.txt">Table of n, a(n) for n = 1..1000</a>
%H A120337 Reijo Ernvall, <a href="http://www.acadsci.fi/mathematica/Vol01/vol01pp195-198.pdf">On the distribution mod 8 of the E-irregular primes</a>, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, Vol. 1, 1975, pp. 195-198.
%H A120337 Reijo Ernvall and Tauno Metsänkylä, <a href="https://doi.org/10.1090/S0025-5718-1978-0482273-9">Cyclotomic invariants and E-irregular primes</a>, Mathematics of Computation, Vol. 32, No. 142 (1978), pp. 617-629; <a href="https://doi.org/10.1090/S0025-5718-1979-0514840-9">Corrigenda</a>, ibid., Vol. 33, No. 145 (1979), p. 433.
%H A120337 Su Hu and Min-Soo Kim, <a href="http://arxiv.org/abs/1510.01558">A note on the irregular primes with respect to Euler polynomials</a>, arXiv:1510.01558 [math.NT], 2015.
%H A120337 Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, <a href="https://arxiv.org/abs/1809.08431">Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture</a>, arXiv:1809.08431 [math.NT], 2018.
%H A120337 Romeo Mestrovic, <a href="http://arxiv.org/abs/1212.3602">A search for primes p such that Euler number E_{p-3} is divisible by p</a>, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 25 2013
%H A120337 Prime Pages, <a href="https://t5k.org/top20/page.php?id=25">Euler Irregular</a>
%H A120337 Samuel S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/full.pdf">Prime divisors of the Bernoulli and Euler numbers</a>, Number theory for the millennium, III, 2002, pp. 357-374, 2002. MR 1956285.
%F A120337 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).
%e A120337 a(1) = 19 because 19 divides E(10) = -19*2659 and 10 + 1 < 19.
%p A120337 A120337_list := proc(bound)
%p A120337 local ae, F, p, m, maxp; F := NULL;
%p A120337 for m from 2 by 2 to bound do
%p A120337   p := nextprime(m+1);
%p A120337   ae := abs(euler(m));
%p A120337   maxp := min(ae, bound);
%p A120337   while p <= maxp do
%p A120337       if ae mod p = 0
%p A120337       then F := F,p fi;
%p A120337       p := nextprime(p);
%p A120337    od;
%p A120337 od;
%p A120337 sort([F]) end: # _Peter Luschny_, Apr 25 2011
%t A120337 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 *)
%Y A120337 Cf. A000928, A092218, A120115, A122045, A290506.
%K A120337 nonn
%O A120337 1,1
%A A120337 _Stefan Krämer_, Jun 22 2006
%E A120337 Terms 251 through 983 from _Peter Luschny_, Apr 25 2011