A120337 -id:A120337 - cp's OEIS frontend

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

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

A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 50521, 691, 41581, 1, 199360981, 3617, 228135437, 43867, 2404879675441, 174611, 14814847529501, 77683, 69348874393137901, 236364091, 238685140977801337, 657931, 4087072509293123892361, 3392780147, 454540704683713199807

Offset: 0

Eric Chen, Dec 15 2014

nonn

Notice: Not all a(n) are 1 or primes, the first example is a(11) = 50521, it equals 19*2659.
a(2n) is a product of powers of Bernoulli irregular primes (A000928), with the exception of n = 0,1,2,3,4,5,7.
a(2n+1) is a product of powers of Euler irregular primes (A120337), with the exception of n = 0,1,2.
Conjectures: All terms are squarefree, and there are infinitely many n such that a(n) is prime.
a(n) = 1 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14}.
a(n) is prime for n = {7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, ...}.
All prime factors of a(n) are irregular primes (Bernoulli or Euler) and with an irregular pair to n: (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ...
Number of ns such that a prime p divides a(n) is the irregular index of p, for example, 67 divides both a(27) and a(58), so it has irregular index two.
a(149) is the first a(n) which is not completely factored (with a 202-digit composite remaining).

Cf. A246006.
Cf. A000928, A120337.
Cf. A001067, A120082, A141590, A060054, A091912.

b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
a[0] = 1; a[n_] := If[EvenQ[n], b[n/2] // Abs, c[(n-1)/2]];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2019 *)

a(2n) = |A001067(n)| = |A120082(2n)| = |A141590(n)| = |A060054(n)|.

a(2n+1) = A091912(n).

A128197 Strong irregular primes. A prime is strong irregular iff it is a Bernoulli irregular prime and an Euler irregular prime.

Original entry on oeis.org

67, 101, 149, 263, 307, 311, 353, 379, 433, 461, 463, 491, 541, 577, 587, 619, 677, 691, 751, 761, 773, 811, 821, 877, 887, 929, 971, 1151, 1229, 1279, 1283, 1291, 1307, 1319, 1381, 1409, 1429, 1439, 1523, 1559, 1621, 1637, 1663, 1733, 1759, 1787, 1831, 1877

Offset: 1

Peter Luschny, Feb 27 2007

Amiram Eldar, Table of n, a(n) for n = 1..424
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.
Peter Luschny, Irregular Bernoulli and Euler Primes.
The Prime Pages, Euler Irregular Primes.
The Prime Pages, Irregular Primes.

Cf. A000040, A000928, A120337.

A250216 Weak irregular primes. A prime is weak irregular iff it is a Bernoulli irregular prime or an Euler irregular prime.

Original entry on oeis.org

19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 359, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 467, 491, 509, 523, 541, 547, 557, 563, 571, 577, 587, 593

Offset: 1

Eric Chen, Dec 24 2014

nonn

Primes p which divide A241601(k) for some k.

Peter Luschny, Irregular Bernoulli and Euler Primes.
The Prime Pages, Irregular Primes
The Prime Pages, Euler Irregular

Cf. A000928, A120337, A128197.

pmax = 593; m0 = 200; dm = 100;
b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
(* a1 = A241601 *) a1[0] = 1; a1[n_] := a1[n] = If[EvenQ[n], b[n/2] // Abs, c[(n - 1)/2]];
f[m_] := f[m] = Module[{}, aa = Table[a1[n], {n, 0, m}]; okQ[p_] := AnyTrue[aa, Divisible[#, p] &]; Reap[For[p = 2, p <= pmax, p = NextPrime[p], If[okQ[p], Sow[p]]]][[2, 1]]];
f[m = m0]; f[m = m + dm];
While[Print["m = ", m]; f[m] != f[m - dm], m = m + dm];
A250216 = f[m] (* Jean-François Alcover, Jul 23 2019 *)

A321217 Genocchi irregular primes.

Original entry on oeis.org

17, 31, 37, 41, 43, 59, 67, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 149, 151, 157, 193, 223, 229, 233, 241, 251, 257, 263, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337, 347, 353, 379, 389, 397, 401, 409, 421, 431, 433, 439, 449, 457, 461, 463, 467, 491, 499

Offset: 1

Michel Marcus, Oct 31 2018

nonn

An odd prime p is G-irregular if it divides at least one of the integers G2, G4, ..., G(p-3).

Conjecture (Hu et al., 2019): The asymptotic density of this sequence within the primes is 1 - 3*A/(2*sqrt(e)) = 0.659776..., where A is Artin's constant (A005596). - Amiram Eldar, Dec 06 2022

Amiram Eldar, Table of n, a(n) for n = 1..1024
Su Hu, Min-Soo Kim, Pieter Moree, and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, Journal of Number Theory, Vol. 205 (2019), pp. 59-80, preprint, arXiv:1809.08431 [math.NT], 2019.
Peter Luschny, Irregular Bernoulli and Euler Primes.
Pieter Moree and Min Sha, Primes in arithmetic progressions and nonprimitive roots, Bulletin of the Australian Mathematical Society (2019), preprint, arXiv:1901.02650 [math.NT], 2019.
Pieter Moree and Pietro Sgobba, Prime divisors of l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l, arXiv:2209.08047 [math.NT], 2022.

Cf. A036968 (Genocchi numbers), A000928 (irregular primes), A120337 (Euler-irregular primes), A128197 (strong irregular primes), A250216 (weak irregular primes), A005596.

A321217_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(m*euler(m-1, 0));
      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: A321217_list(500); # Peter Luschny, Nov 11 2018

G[n_] := G[n] = n EulerE[n - 1, 0];
GenocchiIrregularQ[p_] := AnyTrue[Table[G[k], {k, 2, p-3, 2}], Divisible[#, p]&];
Select[Prime[Range[2, 100]], GenocchiIrregularQ] (* Jean-François Alcover, Nov 16 2018 *)

More terms from Peter Luschny, Nov 11 2018

A250214 Number of values of k such that prime(n) divides A241601(k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 1, 2, 0, 1, 1, 0, 1, 1, 3, 3, 0, 0, 0, 0, 1, 2, 3, 1, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 4, 0, 0, 1, 0, 1, 2, 2, 1, 2, 0, 1, 3, 3, 1, 0, 0, 1, 1, 3, 3, 2, 0, 0, 3, 1, 1

Offset: 1

Eric Chen, Dec 26 2014

nonn

a(n) is called the weak irregular index of n-th prime, that is, the Bernoulli irregular index + Euler irregular index.
Prime(n) is a regular prime if and only if a(n) = 0.
Does every natural number appear in this sequence? For example, for the primes 491 and 1151, a(94) = a(190) = 4. (491 and 1151 are the only primes below 1800 with weak irregular index 4 or more.) However, does a(n) have a limit?

			a(8) = 1 since the 8th prime is 19, which divides A241601(11).
a(13) = 0 since the 13th prime is 41, a regular prime.
a(19) = 2 since the 19th prime is 67, which divides both A241601(27) and A241601(58).

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A091888, A091887, A250216, A000928, A120337, A128197, A250213.

A308243 Index of E-irregularity of prime(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 3, 1, 0, 1, 2, 2, 1, 1, 0, 1, 0, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1

Offset: 1

Felix Fröhlich, May 16 2019

nonn

A prime p >= 5 is an E-irregular prime if there is an even integer 2*k such that 2 <= 2*k <= p-3 and p divides E(2*k), where E(i) is the i-th Euler number (A000364). The pair (p, 2*k) is called an E-irregular pair. The number of such pairs for a given p is called the index of E-irregularity of p (cf. Ernvall, Metsänkylä, 1978, p. 618).

In other words, a prime p is E-irregular if its index of E-irregularity is > 0, which is the case if p is a term of A092218. Otherwise, p is E-regular and is a term of A092217.

R. Ernvall and T. Metsänkylä, Cyclotomic invariants and E-irregular primes, Mathematics of Computation 32 (1978), 617-629.

Cf. A000364, A092217, A092218, A120337.

a000364(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1) \\ after Charles R Greathouse IV in A000364
a(n) = my(p=prime(n), e=2, i=0); while(e <= p-3, if(a000364(e)%p==0, i++); e=e+2); i

A328896 Primes p such that p divides at least one of the integers Fibonacci(2k) for 2k <= p-1.

Original entry on oeis.org

11, 19, 29, 31, 41, 47, 59, 61, 71, 79, 89, 101, 107, 109, 113, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 233, 239, 241, 251, 263, 269, 271, 281, 307, 311, 331, 347, 349, 353, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509

Offset: 1

Felix Fröhlich, Oct 30 2019

nonn

Is the sequence infinite?

Yes, it contains all primes p == 1 or 4 (mod 5), because such p divide Fibonacci(p-1). - Robert Israel, Nov 05 2019

			There are two integers k with 2*k <= 29-1 such that 29 divides Fibonacci(2*k), namely k = 7 and 14, so 29 is a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A000928, A120337, A321217.

filter:= proc(p) local f,k,a,b,t;
a:= -1; b:= 0;
for k from 1 to (p-1)/2 do
   t:= a+2*b mod p;
   a:= a+b mod p; b:= t;
   if t = 0 then return true fi;
od;
false
end proc:
select(filter, [seq(ithprime(i),i=2..100)]); # Robert Israel, Nov 05 2019

forprime(p=1, 100, for(k=1, (p-1)/2, if(Mod(fibonacci(2*k), p)==0, print1(p, ", "); break)))

def isA328896(p):
    return any(p.divides(fibonacci(2*k)) for k in (1..(p-1)//2))
print([p for p in primes(1,510) if isA328896(p)]) # Peter Luschny, Nov 01 2019

Definition corrected by Robert Israel, Nov 05 2019

A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

Original entry on oeis.org

11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, 133, 84, 211, 127, 164, 100, 84, 9, 20, 156, 88, 87, 280, 19, 71, 125, 163, 100, 200, 382, 126, 159, 240, 215, 196, 130, 94, 292, 141, 400, 86, 270, 222, 175, 389, 52, 45, 22, 592, 522, 20

Offset: 1

Eric Chen, Dec 26 2014

nonn

A prime p can divide A241601(k) for more than one k; the first few examples are as follows:
p      k
   27, 58
  63, 68
  130, 147
  62, 110
  211, 239
  100, 213
  88, 91, 137
  87, 193, 292
  19, 257
  71, 186, 300
etc.

			19 is the first weak irregular prime and divides A241601(11) = 50521, so a(1) = 11.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A035112, A120337, A128197, A250216.

More terms from Jinyuan Wang, Feb 18 2022

cp's OEIS Frontend

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

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A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

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A128197 Strong irregular primes. A prime is strong irregular iff it is a Bernoulli irregular prime and an Euler irregular prime.

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A250216 Weak irregular primes. A prime is weak irregular iff it is a Bernoulli irregular prime or an Euler irregular prime.

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A321217 Genocchi irregular primes.

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A250214 Number of values of k such that prime(n) divides A241601(k).

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A308243 Index of E-irregularity of prime(n).

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A328896 Primes p such that p divides at least one of the integers Fibonacci(2k) for 2k <= p-1.

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A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

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