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A007703 Regular primes.
(Formerly M2411)
11
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A prime p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.

REFERENCES

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.

H. M. Edwards, Fermat's Last Theorem, Springer, 1977.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

C. K. Caldwell, The Prime Glossary, Regular prime

K. Conrad, Fermat's Last Theorem For Regular Primes

F. Luca, A. Pizarro-Madariaga, C. Pomerance, On the counting function of irregular primes, 2014.

O. A. Ivanova, Regular prime number

D. Jao, PlanetMath.org, Regular prime

A. L. Robledo, PlanetMath.org, examples of regular primes

Eric Weisstein's World of Mathematics, Regular Prime

Bernoulli numbers, irregularity index of primes

MATHEMATICA

s = {}; Do[p = Prime@n; k = 1; While[2k <= p - 3 && Mod[Numerator@BernoulliB[2k], p] != 0, k++ ]; If[2k > p - 3, AppendTo[s, p]], {n, 2, 80}]; s (* Robert G. Wilson v Sep 20 2006 *)

PROG

(PARI) is(p)=forstep(k=2, p-3, 2, if(numerator(bernfrac(k))%p==0, return(0))); isprime(p) \\ Charles R Greathouse IV, Feb 25 2014

(Python)

from sympy import prime, isprime, bernoulli

from fractions import Fraction

def ok(n):

    for k in xrange(2, n - 2, 2):

        if Fraction(str(bernoulli(k))).numerator%n==0: return 0

    return isprime(n)

print [n for n in xrange(3, 501) if ok(n)] # Indranil Ghosh, Jun 27 2017, after Charles R Greathouse IV

CROSSREFS

Cf. A000928 (irregular primes) and A061576 for further references.

Sequence in context: A165255 A223036 A155058 * A002556 A130101 A130057

Adjacent sequences:  A007700 A007701 A007702 * A007704 A007705 A007706

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

Corrected by Gerard Schildberger, Jun 01 2004

STATUS

approved

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Last modified December 15 03:52 EST 2018. Contains 318141 sequences. (Running on oeis4.)