%I A007703 M2411 #51 Apr 22 2025 10:47:58
%S A007703 3,5,7,11,13,17,19,23,29,31,41,43,47,53,61,71,73,79,83,89,97,107,109,
%T A007703 113,127,137,139,151,163,167,173,179,181,191,193,197,199,211,223,227,
%U A007703 229,239,241,251,269,277,281,313,317,331,337,349,359,367,373,383,397,419,431
%N A007703 Regular primes.
%C A007703 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.
%D A007703 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
%D A007703 H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
%D A007703 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 223-227.
%D A007703 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007703 T. D. Noe, <a href="/A007703/b007703.txt">Table of n, a(n) for n = 1..10000</a>
%H A007703 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/Regular.html">Regular prime</a>.
%H A007703 K. Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf">Fermat's Last Theorem For Regular Primes</a>.
%H A007703 Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H A007703 F. Luca, A. Pizarro-Madariaga, and C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/irreg.pdf">On the counting function of irregular primes</a>, 2014.
%H A007703 O. A. Ivanova, <a href="http://eom.springer.de/R/r080800.htm">Regular prime number</a>.
%H A007703 D. Jao, PlanetMath.org, <a href="https://planetmath.org/regularprime">Regular prime</a>.
%H A007703 A. L. Robledo, PlanetMath.org, <a href="https://planetmath.org/examplesofregularprimes">Examples of regular primes</a>.
%H A007703 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RegularPrime.html">Regular Prime</a>.
%H A007703 <a href="/index/Be#Bernoulli">Bernoulli numbers, irregularity index of primes</a>
%t A007703 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 *)
%o A007703 (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
%o A007703 (Python)
%o A007703 from sympy import prime, isprime, bernoulli
%o A007703 def ok(n):
%o A007703     for k in range(2, n - 2, 2):
%o A007703         if bernoulli(k).as_numer_denom()[0] % n == 0:
%o A007703             return 0
%o A007703     return isprime(n)
%o A007703 [n for n in range(3, 501) if ok(n)] # _Indranil Ghosh_, Jun 27 2017, after _Charles R Greathouse IV_
%Y A007703 Cf. A000928 (irregular primes) and A061576 for further references.
%K A007703 nonn,nice
%O A007703 1,1
%A A007703 _N. J. A. Sloane_, _Simon Plouffe_
%E A007703 Corrected by _Gerard Schildberger_, Jun 01 2004