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%I M5260 N2292 #116 Apr 03 2023 10:36:09
%S 37,59,67,101,103,131,149,157,233,257,263,271,283,293,307,311,347,353,
%T 379,389,401,409,421,433,461,463,467,491,523,541,547,557,577,587,593,
%U 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 Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p.
%C A prime is irregular if and only if the integer Sum_{j=1..p-1} cot^(r)(j*Pi/p)*cot(j*Pi/p) is divisible by p for some even r <= p-5. (See G. Almkvist 1994.) - _Peter Luschny_, Jun 24 2012
%C Jensen proved in 1915 that there are infinitely many irregular primes. It is not known if there are infinitely many regular primes.
%C "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]
%C Johnson (1975) mentions "consecutive irregular prime pairs", meaning an irregular prime p such that, for some integer k <= 2*p-3, p divides the numerators of the Bernoulli numbers B_{2k} and B_{2k+2}. He gives the examples p = 491 (with k=168) and p = 587. No other examples are known. - _N. J. A. Sloane_, May 01 2021, following a suggestion from _Felix Fröhlich_.
%C An odd prime p is irregular if and only if p divides the class number of Q(zeta_p), where zeta_n = exp(2*Pi*i/n); that is, for k >= 2, p = prime(k) is irregular if and only if p divides A055513(k). For example, 37 is irregular since Q(zeta_37) has class number A055513(12) = 37. - _Jianing Song_, Sep 13 2022
%D G. Almkvist, Wilf's conjecture and a generalization, In: The Rademacher legacy to mathematics, 211-233, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, 1994.
%D Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 377, 425-430 (but there are errors in the tables).
%D R. E. Crandall, Mathematica for the Sciences, Addison-Wesley Publishing Co., Redwood City, CA, 1991, pp. 248-255.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 59, p. 21, Ellipses, Paris 2008.
%D H. M. Edwards, Fermat's Last Theorem, Springer, 1977, see p. 244.
%D J. Neukirch, Algebraic Number Theory, Springer, 1999, p. 38.
%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 257.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.
%H T. D. Noe, <a href="/A000928/b000928.txt">Table of n, a(n) for n = 1..10000</a>
%H Abiessu, <a href="http://everything2.net/index.pl?node_id=1214159&displaytype=printable&lastnode_id=1214159">Irregular prime</a>
%H C. Banderier, <a href="http://algo.inria.fr/banderier/Recipro/node38.html">Nombres premiers réguliers</a> (in French).
%H 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 J. P. Buhler, R. E. Crandall and R. W. Sompolski, <a href="https://doi.org/10.1090/S0025-5718-1992-1134717-4">Irregular primes to one million</a>, Math. Comp. 59 no 200 (1992) 717-722.
%H Joe P. Buhler and David Harvey, <a href="http://www.cims.nyu.edu/~harvey/irregular/">Irregular primes to 163 million</a>
%H Joe P. Buhler and David Harvey, <a href="http://arxiv.org/abs/0912.2121">Irregular primes to 163 million</a>, arXiv:0912.2121 [math.NT], 2009.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/Regular.html">Regular prime</a>
%H C. K. Caldwell, the top twenty, <a href="https://t5k.org/top20/page.php?id=26">Irregular Primes</a>
%H V. A. Demyanenko, <a href="http://eom.springer.de/I/i052670.htm">Irregular prime number</a>
%H G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Ward/ward2.html">Integer Sequences and Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3.
%H William Hart, David Harvey and Wilson Ong, <a href="https://doi.org/10.1090/mcom/3211">Irregular primes to two billion</a>, Math. Comp. 86 (2017), 3031-3049. <a href="https://arxiv.org/abs/1605.02398">Preprint arXiv:1605.02398 [math.NT]</a>.
%H David Harvey, <a href="https://web.maths.unsw.edu.au/~davidharvey/research/twobillion.tar.bz2">List of irregular pairs</a>, 2017. Download size 557 MB.
%H Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, <a href="https://arxiv.org/abs/1809.08431">Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture</a>, arXiv:1809.08431 [math.NT], 2018.
%H K. L. Jensen, <a href="https://www.jstor.org/stable/24532219">Om talteoretiske Egenskaber ved de Bernoulliske Tal</a>, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.
%H W. Johnson, <a href="http://dx.doi.org/10.1090/S0025-5718-1973-0384748-5">On the vanishing of the Iwasawa invariant {mu}_p for p < 8000</a>, Math. Comp., 27 (1973), 387-396 (gives a list up to 8000 and points out that 1381, 1597, 1663, 1877 were omitted from earlier lists).
%H W. Johnson, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0347727-0">Irregular prime divisors of the Bernoulli numbers</a>, Math. Comp. 28 (1974), 653-657.
%H W. Johnson, <a href="https://doi.org/10.1090/S0025-5718-1975-0376606-9">Irregular primes and cyclotomic invariants</a>, Math. Comp. 29 (1975), 113-120.
%H B. C. Kellner, <a href="https://doi.org/10.1090/S0025-5718-06-01887-4">On Irregular Prime Power Divisors of the Bernoulli Numbers</a>, Math. Comp. 75 (2006) PII S0025-5718(06)01887-4.
%H 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>, Proc. Nat. Acad. Sci. USA, 40 (1954), 25-33 (but there are errors).
%H C. Lin and L. Zhipeng, <a href="https://arxiv.org/abs/math/0408082">On Bernoulli numbers and its properties</a>, arXiv:math/0408082 [math.HO], 2004.
%H 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 Peter Luschny, <a href="http://www.luschny.de/math/primes/irregular.html">The Computation of Irregular Primes.</a> [From _Peter Luschny_, Apr 20 2009]
%H R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%H Tauno Metsänkylä, <a href="https://doi.org/10.5186/aasfm.1971.492">Note on the distribution of irregular primes</a>, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, 492, 1971.
%H 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 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 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 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 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 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 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 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 S. S. Wagstaff, Jr, <a href="https://doi.org/10.1090/S0025-5718-1978-0491465-4">The Irregular Primes to 125000</a>, Math. Comp. 32 no 142 (1978) 583-592.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrregularPrime.html">Irregular Prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H <a href="/index/Be#Bernoulli">Bernoulli numbers, irregularity index of primes</a>
%p A000928_list := proc(len)
%p local ab, m, F, p, maxp; F := {};
%p for m from 2 by 2 to len do
%p p := nextprime(m+1);
%p ab := abs(bernoulli(m));
%p maxp := min(ab, len);
%p while p <= maxp do
%p if ab mod p = 0
%p then F := F union {p} fi;
%p p := nextprime(p);
%p od;
%p od;
%p sort(convert(F,list)) end:
%p A000928_list(1000); # _Peter Luschny_, Apr 25 2011
%t fQ[p_] := Block[{k = 1}, While[ 2k <= p-3 && Mod[ Numerator@ BernoulliB[ 2k], p] != 0, k++]; 2k <= p-3]; Select[ Prime@ Range@ 137, fQ] (* _Robert G. Wilson v_, Jun 25 2012 *)
%t Select[Prime[Range[200]],MemberQ[Mod[Numerator[BernoulliB[2*Range[(#-1)/ 2]]], #],0]&] (* _Harvey P. Dale_, Mar 02 2018 *)
%o (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 */
%o (Python)
%o from sympy import bernoulli, primerange
%o def ok(n):
%o k = 1
%o while 2*k <= n - 3 and bernoulli(2*k).numerator() % n:
%o k+=1
%o return 2*k <= n - 3
%o print([n for n in primerange(2, 1101) if ok(n)]) # _Indranil Ghosh_, Jun 27 2017, after _Robert G. Wilson v_
%Y Cf. A007703, A061576.
%Y Cf. A091887 (irregularity index of the n-th irregular prime).
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
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