A092219 -id:A092219 - cp's OEIS frontend

A092217 Primes that do not divide any Euler number.

2, 3, 7, 11, 23, 59, 83, 103, 107, 127, 131, 151, 163, 167, 179, 191, 199, 211, 227, 239, 271, 283, 331, 347, 367, 383, 431, 439, 443, 467, 479, 487, 499, 503, 523, 547, 599, 607, 631, 643, 647, 659, 683, 719, 727, 743, 787, 823, 827, 839, 859, 863, 883, 911

Offset: 1

T. D. Noe, Feb 25 2004

nonn

After computing the Euler numbers, finding the non-divisors is simple because the Euler numbers satisfy a Kummer congruence. See Wagstaff for details. The density of these primes is approximately 0.33.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..264 from T. D. Noe)
Samuel S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers, in: Number Theory for the Millennium III, A K Peters, 2002, pp. 357-374.
Eric Weisstein's World of Mathematics, Euler Number

Cf. A000364 (Euler numbers), A092218 (primes that divide some Euler number), A092219.

ee=Table[Abs[EulerE[2i]], {i, 1000}]; t=Table[p=Prime[n]; cnt=0; Do[If[Mod[ee[[i]], p]==0, cnt++ ], {i, p}]; cnt, {n, PrimePi[1000]}]; Prime[Flatten[Position[t, 0]]]

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&]]

cp's OEIS Frontend

A092217 Primes that do not divide any Euler number.

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