A092217 -id:A092217 - cp's OEIS frontend

A092218 Primes that divide some Euler number.

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

A092219 Least even k such that the Euler number E(k) is divisible by prime(n), or 0 if there is no k.

Original entry on oeis.org

0, 0, 4, 0, 0, 12, 16, 10, 0, 28, 22, 36, 40, 12, 14, 52, 0, 6, 26, 28, 72, 18, 0, 88, 96, 62, 0, 0, 108, 112, 0, 0, 42, 128, 146, 0, 156, 0, 0, 172, 0, 180, 0, 74, 196, 0, 0, 132, 0, 228, 232, 0, 210, 126, 256, 212, 268, 0, 8, 280, 0, 292, 90, 86, 312, 316, 0, 336, 0, 18, 70

Offset: 1

T. D. Noe, Feb 25 2004

nonn

A prime p will either divide an Euler number E(k) for k < p or divide no Euler number. This sequence can be used to find A092218, primes that divide Euler numbers and A092217, primes that divide no Euler number.

T. D. Noe, Table of n, a(n) for n = 1..850
S. S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers
Eric Weisstein's World of Mathematics, Euler Number

Cf. A000364 (Euler numbers).

ee=Table[Abs[EulerE[2i]], {i, 500}]; Table[p=Prime[n]; i=1; While[i<=p && Mod[ee[[i]], p]>0, i++ ]; If[i>p, 0, 2i], {n, PrimePi[500]}]

A140111 Primes of the form prime(x) + prime(x+1) - (prime(x+1) - prime(x))/2.

Original entry on oeis.org

7, 11, 23, 59, 83, 97, 109, 149, 233, 283, 317, 337, 349, 359, 367, 383, 479, 487, 563, 571, 593, 641, 709, 769, 823, 839, 863, 1009, 1117, 1129, 1217, 1283, 1297, 1319, 1423, 1553, 1579, 1619, 1627, 1663, 1733, 1913, 2029, 2039, 2063, 2069, 2083, 2099, 2129

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 21 2008

nonn

			3+5-(5-3)/2=3+5-1=7, 5+7-(7-5)/2=5+7-1=11, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A092217.

a = {}; Do[p1 = Prime[n]; p2 = Prime[n + 1]; p = p1 + p2 - (p2 - p1)/2; If[PrimeQ[p], AppendTo[a, p]], {n, 10^2}]; a
Select[#[[1]]+#[[2]]-(#[[2]]-#[[1]])/2&/@Partition[ Prime[ Range[ 200]],2,1],PrimeQ] (* Harvey P. Dale, Mar 14 2021 *)

More terms from Harvey P. Dale, Mar 14 2021

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

A308270 Smallest prime with E-irregularity index n.

Original entry on oeis.org

2, 5, 307, 277

Offset: 0

Felix Fröhlich, May 17 2019

Smallest prime p such that A308243(i) = n, where i is the index of p in A000040.
E-regular primes have E-irregularity index 0, so a(0) = 2, since 2 is the smallest E-regular prime (A092217).
Does such a prime exist for every n?
a(4) > 2003 if it exists.

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

Cf. A000040, A061576, A092217, A308243.

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
a308243(n) = my(p=prime(n), e=2, i=0); while(e <= p-3, if(a000364(e)%p==0, i++); e=e+2); i
a(n) = for(x=1, oo, if(a308243(x)==n, return(prime(x))))

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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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A092219 Least even k such that the Euler number E(k) is divisible by prime(n), or 0 if there is no k.

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A140111 Primes of the form prime(x) + prime(x+1) - (prime(x+1) - prime(x))/2.

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

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A308270 Smallest prime with E-irregularity index n.

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