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A000928
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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)
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73
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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
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OFFSET
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1,1
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COMMENTS
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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]
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.
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
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REFERENCES
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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.
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.
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LINKS
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MAPLE
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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:
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MATHEMATICA
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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[200]], MemberQ[Mod[Numerator[BernoulliB[2*Range[(#-1)/ 2]]], #], 0]&] (* Harvey P. Dale, Mar 02 2018 *)
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PROG
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(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
def ok(n):
k = 1
while 2*k <= n - 3 and bernoulli(2*k).numerator() % n:
k+=1
return 2*k <= n - 3
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CROSSREFS
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Cf. A091887 (irregularity index of the n-th irregular prime).
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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