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 A000928 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. (Formerly M5260 N2292) 71
 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 Jensen proved in 1915 that there are infinitely many irregular primes. It is not known if there are infinitely many regular primes. "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] REFERENCES 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. Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 377, 425-430 (but there are errors in the tables). R. E. Crandall, Mathematica for the Sciences, Addison-Wesley Publishing Co., Redwood City, CA, 1991, pp. 248-255. J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 59, p. 21, Ellipses, Paris 2008. H. M. Edwards, Fermat's Last Theorem, Springer, 1977, see p. 244. Jensen, K. L. "Om talteoretiske Egenskaber ved de Bernoulliske Tal." Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83. J. Neukirch, Algebraic Number Theory, Springer, 1999, p. 38. P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 257. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 Abiessu, Irregular prime C. Banderier, Nombres premiers réguliers (in French). J. P. Buhler, R. E. Crandall, R. Ernvall et al., Irregular primes and cyclotomic invariants to 12 Million,J. Symbolic Computation 31 (2001) 89-96. J. P. Buhler, R. E. Crandall and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 no 200 (1992) 717-722. Joe P. Buhler and David Harvey, Irregular primes to 163 million Joe P. Buhler and David Harvey, Irregular primes to 163 million, arXiv:0912.2121 [math.NT], 2009. C. K. Caldwell, The Prime Glossary, Regular prime C. K. Caldwell, the top twenty, Irregular Primes V. A. Demyanenko, Irregular prime number G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3. Su Hu, Min-Soo Kim, Pieter Moree, Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018. W. Johnson, On the vanishing of the Iwasawa invariant {mu}_p for p < 8000, 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). W. Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653-657. B. C. Kellner, On Irregular Prime Power Divisors of the Bernoulli Numbers, Math. Comp. 75 (2006) PII S0025-5718(06)01887-4 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). C. Lin and L. Zhipeng, On Bernoulli numbers and its properties, arXiv:math/0408082 [math.HO], 2004. F. Luca, A. Pizarro-Madariaga, C. Pomerance, On the counting function of irregular primes, 2014. Peter Luschny, The Computation of Irregular Primes. [From Peter Luschny, Apr 20 2009] R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4. H. S. Vandiver, Note On The Divisors Of The Numerators Of Bernoulli's Numbers H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem S. S. Wagstaff, Jr, The Irregular Primes to 125000, Math. Comp. 32 no 142 (1978) 583-592. Eric Weisstein's World of Mathematics, Irregular Prime Eric Weisstein's World of Mathematics, Integer Sequence Primes MAPLE A000928_list := proc(len) local ab, m, F, p, maxp; F := {}; for m from 2 by 2 to len do    p := nextprime(m+1);    ab := abs(bernoulli(m));    maxp := min(ab, len);    while p <= maxp do       if ab mod p = 0       then F := F union {p} fi;       p := nextprime(p);    od; od; sort(convert(F, list)) end: A000928_list(1000); # Peter Luschny, Apr 25 2011 MATHEMATICA 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 *) Select[Prime[Range], MemberQ[Mod[Numerator[BernoulliB[2*Range[(#-1)/ 2]]], #], 0]&] (* Harvey P. Dale, Mar 02 2018 *) PROG (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 */ (Python) from sympy import bernoulli, primerange from fractions import Fraction def ok(n):     k=1     while 2*k<=n - 3 and Fraction(str(bernoulli(2*k))).numerator%n!=0: k+=1     return 2*k<=n - 3 print [n for n in primerange(2, 1101) if ok(n)] # Indranil Ghosh, Jun 27 2017, after Robert G. Wilson v CROSSREFS Cf. A007703, A061576. Cf. A091887 (irregularity index of the n-th irregular prime). Sequence in context: A127023 A109166 A090798 * A073276 A281290 A105460 Adjacent sequences:  A000925 A000926 A000927 * A000929 A000930 A000931 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)