A120115 -id:A120115 - cp's OEIS frontend

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

Stefan Krämer, Jun 22 2006

nonn

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

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

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.

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

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

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 *)

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).

Terms 251 through 983 from Peter Luschny, Apr 25 2011

A092218 Primes that divide some Euler number.

Original entry on oeis.org

5, 13, 17, 19, 29, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 89, 97, 101, 109, 113, 137, 139, 149, 157, 173, 181, 193, 197, 223, 229, 233, 241, 251, 257, 263, 269, 277, 281, 293, 307, 311, 313, 317, 337, 349, 353, 359, 373, 379, 389, 397, 401, 409, 419, 421

Offset: 1

T. D. Noe, Feb 25 2004

nonn

For a prime p in this sequence, p will divide an Euler number E(k) for k < p. The density of these primes is approximately 0.66.

This sequence is the union of A002144 (primes of the form 4k+1) and A120115. Note that if prime p=1 (mod 4), then p divides E(p-1). - T. D. Noe, Jun 09 2006

T. D. Noe, Table of n, a(n) for n = 1..586
L. Carlitz, Note on irregular primes, Proc. Amer. Math. Soc. 5 (1954), 329-331
S. S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers
Eric Weisstein's World of Mathematics, Euler Number

Cf. A000364 (Euler numbers), A092217 (primes that do not divide any Euler number), A092219.

ee=Table[Abs[EulerE[2i]], {i, 500}]; t=Table[p=Prime[n]; cnt=0; Do[If[Mod[ee[[i]], p]==0, cnt++ ], {i, p}]; cnt, {n, PrimePi[500]}]; Prime[Select[Range[Length[t]], t[[ # ]]>0&]]

A198245 Euler primes: Primes p that divide E(p - 3), where E(k) is the k-th Euler number.

Original entry on oeis.org

149, 241, 2946901, 16467631, 17613227, 327784727, 426369739, 1062232319

Offset: 1

Romeo Mestrovic, Oct 22 2011

Also called Vandiver primes. - N. J. A. Sloane, Sep 25 2023
See A196230 for another sequence of "Euler primes". - N. J. A. Sloane, May 29 2022
The even-indexed Euler numbers are A028296, the odd-indexed Euler numbers are all zero.
Numerous combinatorial congruences recently obtained by Z. W. Sun and by Z. H. Sun contain the Euler numbers E(p-3) with a prime p.
Only three primes less than 3 * 10^6 satisfy this condition (the current members of the sequence).
Such primes have been recently suggested by Z. W. Sun; namely, Sun found the first and the second such primes, 149 and 241, and used them to discover new congruences involving E(p - 3).
This is reported by Zhi Wei Sun on Feb 08 2010 and the third prime was found by Romeo Mestrovic (on Sep 26 2011).
Mestrovic (2012) computes that only three primes < 10^7 are in the sequence, but he conjectures that the sequence is infinite. - Jonathan Sondow, Dec 18 2012
If it exists, a(9) > 2 * 10^9. - Hiroaki Yamanouchi, Aug 06 2017
Hathi et al. give a(3) as 2124679 and claim that the terms 2124679, 16467631, 17613227 were reported in Cosgrave, Dilcher, 2013, but 2124679 does not appear in table 2 in that paper. How is 2124679 related to this sequence? Note that 2124679 is the second Wolstenholme prime (A088164). - Felix Fröhlich, Apr 27 2021

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.8.

A. R. Booker, S. Hathi, M. J. Mossinghoff and T. S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, 58 (2022), 913-941, arXiv:2101.11157. See Theorem 1, but beware errors.
John B. Cosgrave and Karl Dilcher, On a congruence of Emma Lehmer related to Euler numbers, Acta Arithmetica 161 (2013), 47-67.
Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, Wolstenholme and Vandiver primes, arXiv:2101.11157 [math.NT], 2021.
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. vol 76, no 260 (2007) pp 2087-2094.
Romeo Mestrovic, An extension of a congruence by Kohnen, arXiv: 1109.2340v3 [math.NT] (2011).
Romeo Mestrovic, A search for primes p such that Euler number E(p-3) is divisible by p, arXiv: 1212.3602 [math.NT] (2012).
Zhi Wei Sun, Letter to the Number Theory List, Feb 8 2010
Zhi Wei Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), in press, arXiv: 1001.4453 [math.NT] (2011).
Eric Weisstein's World of Mathematics, Euler Number.
Wikipedia, Euler Number.
Hiroaki Yamanouchi, Primes p (5 <= p < 2*10^9) such that E(p-3) == A (mod p) for some integer A in [-1000, 1000].

Cf. A000364, A088164, A092217, A092218, A120115, A120337, A196230.

Select[Prime[Range[2, 200]], IntegerQ[EulerE[# - 3]/#] &] (* Alonso del Arte, Oct 31 2011 *)

a(4)-a(8) from Hiroaki Yamanouchi, Aug 06 2017

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

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A092218 Primes that divide some Euler number.

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A198245 Euler primes: Primes p that divide E(p - 3), where E(k) is the k-th Euler number.

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