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A120337 Euler-irregular primes p dividing E(2k) for some 2k<p-1. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

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

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

R. Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602, 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. A092218, A120115.

Sequence in context: A178251 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

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.