

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
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OFFSET

1,1


COMMENTS

A prime p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p3} (A000367) are not divisible by p.


REFERENCES

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425430.
H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
F. Luca, A. PizarroMadariaga, C. Pomerance, On the counting function of irregular primes, https://math.dartmouth.edu/~carlp/irreg.pdf, 2014.
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
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, p3, 2, if(numerator(bernfrac(k))%p==0, return(0))); isprime(p) \\ Charles R Greathouse IV, Feb 25 2014


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



