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Euler-irregular primes p dividing E(2k) for some 2k < p-1.
10

%I #40 Apr 03 2023 10:36:11

%S 19,31,43,47,61,67,71,79,101,137,139,149,193,223,241,251,263,277,307,

%T 311,349,353,359,373,379,419,433,461,463,491,509,541,563,571,577,587,

%U 619,677,691,709,739,751,761,769,773,811,821,877,887,907,929,941,967,971,983

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

%C 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 Robert G. Wilson v, <a href="/A120337/b120337.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 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 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 R. 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 Prime Pages, <a href="https://t5k.org/top20/page.php?id=25">Euler Irregular</a>

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

%p A120337_list := proc(bound)

%p local ae, F, p, m, maxp; F := NULL;

%p for m from 2 by 2 to bound do

%p p := nextprime(m+1);

%p ae := abs(euler(m));

%p maxp := min(ae, bound);

%p while p <= maxp do

%p if ae mod p = 0

%p then F := F,p fi;

%p p := nextprime(p);

%p od;

%p od;

%p sort([F]) end: # _Peter Luschny_, Apr 25 2011

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

%K nonn

%O 1,1

%A _Stefan Krämer_, Jun 22 2006

%E Terms 251 through 983 from _Peter Luschny_, Apr 25 2011