%I A000928 M5260 N2292
%S A000928 37,59,67,101,103,131,149,157,233,257,263,271,283,293,307,311,347,
%T A000928 353,379,389,401,409,421,433,461,463,467,491,523,541,547,557,577,587,593,
%U A000928 607,613,617,619,631,647,653,659,673,677,683,691,727,751,757,761,773,797,
               809,811,821,827,839,877,881,887,929,953,971,1061
%N A000928 Irregular primes: 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.
%C A000928 Jensen proved in 1915 that there are infinitely many irregular primes. 
               It is not known if there are infinitely many regular primes.
%C A000928 "The pioneering mathematician Kummer, over the period 1847-1850, used 
               his profound theory of cyclotomic fields to establish a certain class 
               of primes called 'regular' primes. ... It is known that there exist 
               an infinity of irregular primes; in fact it is a plausible conjecture 
               that only an asymptotic fraction 1/Sqrt(e) ~ 0.6 of all primes are 
               regular" [Ribenboim]
%D A000928 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, 
               NY, 1966, pp. 425-430 (but there are errors).
%D A000928 R. E. Crandall, Mathematica for the Sciences, Addison-Wesley Publishing 
               Co., Redwood City, CA, 1991, pp. 248-255.
%D A000928 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 59, p. 21, Ellipses, 
               Paris 2008.
%D A000928 H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
%D A000928 W. Johnson, On the vanishing of the Iwasawa invariant {mu}_p for p < 
               8000, Math. Comp., 27 (1973), 387-396 (points out that 1381, 1597, 
               1663, 1877 were omitted from earlier lists).
%D A000928 W. Johnson, Irregular prime divisors of the Bernoulli numbers, Math. 
               Comp. 28 (1974), 653-657.
%D A000928 D. H. Lehmer et al., An application of high-speed computing to Fermat's 
               last theorem, Proc. Nat. Acad. Sci. USA, 40 (1954), 25-33 (but there 
               are errors).
%D A000928 J. Neukirch, Algebraic Number Theory, Springer, 1999, p. 38.
%D A000928 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 137.
%D A000928 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000928 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000928 L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.
%H A000928 T. D. Noe, <a href="b000928.txt">Table of n, a(n) for n = 1..10000</a>
%H A000928 Abiessu, <a href="http://everything2.net/index.pl?node_id=1214159&displaytype=printable&lastnode_id=1214159">
               Irregular prime</a>
%H A000928 C. Banderier, <a href="http://algo.inria.fr/banderier/Recipro/node38.html">
               Nombres premiers reguliers</a>
%H A000928 J. P. Buhler, R. E. Crandall, R. Ernvall et al., <a href="http://dx.doi.org/
               10.1006/jsco.1999.1011">Irregular primes and cyclotomic invariants 
               to 12 Million</a>,J. Symbolic Computation 31 (2001) 89-96.
%H A000928 J. P. Buhler, R. E. Crandall and R. W. Sompolski, <a href="http://links.jstor.org/
               sici?sici=0025-5718%28199210%2959%3A200%3C717%3AIPTOM%3E2.0.CO%3B2-Q">
               Irregular primes to one million</a>, Math. Comp. 59 no 200 (1992) 
               717-722.
%H A000928 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php/Regular.html">Regular prime</a>
%H A000928 C. K. Caldwell, the top twenty, <a href="http://primes.utm.edu/top20/
               page.php?id=26">Irregular Primes</a>
%H A000928 V. A. Demyanenko, <a href="http://eom.springer.de/I/i052670.htm">Irregular 
               prime number</a>
%H A000928 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http:/
               /www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and 
               Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), 
               Article 02.2.3
%H A000928 B. C. Kellner, <a href="http://www.ams.org/jourcgi/jrnl_toolbar_nav/mcom_all">
               On Irregular Prime Power Divisors of the Bernoulli Numbers</a>, Math. 
               Comp. 75 (2006) PII S0025-5718(06)01887-4
%H A000928 D. H. Lehmer et al., <a href="http://www.pnas.org/cgi/reprint/40/1/25.pdf">
               An Application Of High-Speed Computing To Fermat's Last Theorem</
               a>
%H A000928 C. Lin and L. Zhipeng, <a href="http://arXiv.org/abs/math.HO/0408082">
               On Bernoulli numbers and its properties</a>
%H A000928 Peter Luschny, <a href="http://www.luschny.de/math/primes/irregular.html">
               The Computation of Irregular Primes.</a> [From Peter Luschny (peter(AT)luschny.de), 
               Apr 20 2009]
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/18/9/594.pdf">
               Note On The Divisors Of The Numerators Of Bernoulli's Numbers</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/12/2/106.pdf">
               Summary Of Results And Proofs Concerning Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/12/12/767.pdf">
               Summary Of Results And Proofs Concerning Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/15/1/43.pdf">
               Summary Of Results And Proofs Concerning Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/15/2/108.pdf">
               Summary Of Results And Proofs On Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/16/4/298.pdf">
               Summary Of Results And Proofs On Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/17/12/661.pdf">
               Summary Of Results And Proofs On Fermat's Last Theorem</a>
%H A000928 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/40/8/732.pdf">
               Examination Of Methods Of Attack On The Second Case Of Fermat's Last 
               Theorem</a>
%H A000928 S. S. Wagstaff, Jr, <a href="http://links.jstor.org/sici?sici=0025-5718%28197804%2932%3A142%3C%3E1.0.CO%3B2-U\
               ">The Irregular Primes to 125000</a>, Math. Comp. 32 no 142 (1978) 
               583-592
%H A000928 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IrregularPrime.html">Link to a section of The World of Mathematics.</
               a>
%H A000928 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A000928 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%H A000928 <a href="Sindx_Be.html#Bernoulli">Bernoulli numbers, irregularity index 
               of primes</a>
%t A000928 Do[ p = Prime[ n ]; k = 1; While[ 2*k <= p - 3 && Mod[ Numerator[ BernoulliB[ 
               2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 1, Print[ p ] ], { n, 3, 
               200} ]
%t A000928 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, 137}]; s 
               (* Robert G. Wilson v *)
%o A000928 (PARI) a(n)=local(p);if(n<1,0,p=a(n-1)+(n==1);while(p=nextprime(p+2), 
               forstep(i=2,p-3,2,if(numerator(bernfrac(i))%p==0,break(2))));p) - 
               Michael Somos Feb 04 2004
%Y A000928 Cf. A007703, A061576.
%Y A000928 Cf. A091887 (irregularity index of the n-th irregular prime).
%Y A000928 Sequence in context: A127023 A109166 A090798 this_sequence A073276 A105460 
               A141851
%Y A000928 Adjacent sequences: A000925 A000926 A000927 this_sequence A000929 A000930 
               A000931
%K A000928 nonn,nice,easy
%O A000928 1,1
%A A000928 N. J. A. Sloane (njas(AT)research.att.com).
%E A000928 Johnson (1973) gives a list up to 8000.