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A007703
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Regular primes.
(Formerly M2411)
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12
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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