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A007703 Regular primes.
(Formerly M2411)
12
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
C. K. Caldwell, The Prime Glossary, Regular prime.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
F. Luca, A. Pizarro-Madariaga, and 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.
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
def ok(n):
for k in range(2, n - 2, 2):
if bernoulli(k).as_numer_denom()[0] % n == 0:
return 0
return isprime(n)
[n for n in range(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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Corrected by Gerard Schildberger, Jun 01 2004
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)