A055513 -id:A055513 - cp's OEIS frontend

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.

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

Offset: 1

N. J. A. Sloane

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

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.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 257.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 225.
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.

T. D. Noe, Table of n, a(n) for n = 1..10000
Abiessu, Irregular prime
C. Banderier, Nombres premiers réguliers (in French). [Cached copy at the Wayback Machine]
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.
William Hart, David Harvey and Wilson Ong, Irregular primes to two billion, Math. Comp. 86 (2017), 3031-3049. Preprint arXiv:1605.02398 [math.NT].
David Harvey, List of irregular pairs, 2017. Download size 557 MB.
Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018.
K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.
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.
W. Johnson, Irregular primes and cyclotomic invariants, Math. Comp. 29 (1975), 113-120.
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.
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 and C. Pomerance, On the counting function of irregular primes, 2014.
Peter Luschny, The Computation of Irregular Primes. [From _Peter Luschny_, Apr 20 2009]
Romeo Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Tauno Metsänkylä, Note on the distribution of irregular primes, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, 492, 1971.
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
H. S. Vandiver, Examination Of Methods Of Attack On The Second Case Of 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
Index entries for sequences related to Bernoulli numbers.
Bernoulli numbers, irregularity index of primes

Cf. A007703, A061576.

Cf. A091887 (irregularity index of the n-th irregular prime).

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

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 *)

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 */

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
print([n for n in primerange(2, 1101) if ok(n)]) # Indranil Ghosh, Jun 27 2017, after Robert G. Wilson v

A000927 "First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 8, 9, 37, 121, 211, 695, 4889, 41241, 76301, 853513, 3882809, 11957417, 100146415, 838216959, 13379363737, 411322824001, 3547404378125, 9069094643165, 63434933542623, 161784800122409, 1612072001362952, 2604529186263992195, 28496379729272136525, 646901570175200968153, 1753848916484925681747, 687887859687174720123201, 2333546653547742584439257, 56234327700401832767069245, 2708534744692077051875131636

Offset: 1

N. J. A. Sloane

Washington gives a very extensive table. But beware errors: Washington incorrectly gives a(17) = 41421, a(25) = 411322842001 (corrected in the second edition).

			For n = 9, prime(9) = 23, a(9) = 3.
For n = 38, prime(38) = 163, a(38) = 2708534744692077051875131636.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 429.
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, pp. 353-360 (1st edition) pp. 412-420 (2nd edition).

Max Alekseyev, Table of n, a(n) for n = 1..100
Hisanori Mishima, Factorizations of Cyclotomic Numbers
M. Newman, A table of the first factor for prime cyclotomic fields, Math. Comp., 24 (1970), 215-219.
M. A. Shokrollahi, Tables

Subsequence of A061653.

For the full class number h = h- * h+, see A055513, which agrees for the first 36 terms, assuming the Generalized Riemann Hypothesis.

f:= proc(n) uses LinearAlgebra;
  local p,M;
  p:= ithprime(n);
  M:= Matrix((p-3)/2,(p-3)/2,(i,j) -> floor((i+1)*(j+2)/p) - floor(i*(j+2)/p));
  abs(Determinant(M));
end proc:
1, seq(f(n),n=3..50); # Robert Israel, Sep 20 2016

a[n_]:= With[{p = Prime[n]}, If[n<4, 1, Abs[ Det[ Table[ Quotient[ (i+2)*(j+2), p] - Quotient[ (i+1)*(j+2), p], {i, 1, (p-1)/2-2}, {j, 1, (p-1)/2-2}]]]]]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Aug 01 2013, translated from Pari; modified by G. C. Greubel, Aug 08 2019 *)

{ A000927(n) = if(n<3,return(1)); my(p=prime(n)); abs( matdet(matrix((p-1)/2-2, (p-1)/2-2, i, j, ((i+2)*(j+2))\p - ((i+1)*(j+2))\p)) ); } \\ Max Alekseyev, Oct 31 2012; corrected by G. C. Greubel and Michel Marcus, Aug 07 2019

For n>2, a(n) equals absolute value of determinant of the matrix with entries floor(i*j/p)-floor((i-1)*j/p), 3 <= i,j <= (p-1)/2, where p = prime(n) = A000040(n). - Max Alekseyev, Oct 31 2012

a(n) = A061653(A000040(n)).

Edited by Max Alekseyev, Oct 25 2012

a(1)=1 prepended by Max Alekseyev, Mar 05 2018

A061653 Relative class number h- of cyclotomic field Q(zeta_n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 9, 1, 1, 1, 1, 1, 37, 1, 2, 1, 121, 1, 211, 1, 1, 3, 695, 1, 43, 1, 5, 3, 4889, 1, 10, 2, 9, 8, 41241, 1, 76301, 9, 7, 17, 64, 1, 853513, 8, 69, 1, 3882809, 3, 11957417, 37, 11, 19, 1280, 2, 100146415

Offset: 1

N. J. A. Sloane, Jun 16 2001

Note that if n == 2 (mod 4), Q(zeta_n) is the same field as Q(zeta_{n/2}).
From Richard N. Smith, Jul 15 2019: (Start)
For prime p, p divides a(p) (or a(2p)) if and only if p is in A000928.
For prime p, p divides a(4p) if and only if p is in A250216. (End)

			Q(zeta_23) = 3 is the first time that h- is bigger than 1.

Richard N. Smith, Table of n, a(n) for n = 1..256 (terms 1..163 from R. J. Mathar)
Dylan Johnston, Diego Martín Duro, and Dmitriy Rumynin, Disconnected Reductive Groups: Classification and Representations, arXiv:2409.06375 [math.RT], 2024. See p. 14.
L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353. [WARNING: The table contains errors for n=59, 97, ...]

Contains A000927, A035115, A061494 as subsequences.

Cf. A055513, A005848.

For prime p, a(p) = A000927(A000720(p)).

Washington gives an extensive table on pp. 353-360.
Missing term a(1) = 1 inserted by N. J. A. Sloane, Feb 05 2009 at the suggestion of Tanya Khovanova
More terms from R. J. Mathar, Feb 06 2009
a(59) changed from 41421 to 41241 (given correctly in 2nd edition of Washington), Matthew Johnson, Jul 20 2013
a(59) in b-file changed as above by Andrew Howroyd, Feb 23 2018
a(97) corrected, a(163) added by Max Alekseyev, Mar 05 2018

A088922 Consider the n X n matrix with entries (i*j mod n), where i,j=0..n-1; a(n) = rank of this matrix over the real numbers.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 6, 7, 6, 10, 7, 9, 10, 11, 9, 13, 10, 14, 13, 13, 12, 18, 14, 15, 16, 18, 15, 21, 16, 20, 19, 19, 20, 25, 19, 21, 22, 26, 21, 27, 22, 26, 27, 25, 24, 32, 26, 29, 28, 30, 27, 33, 30, 34, 31, 31, 30, 40, 31, 33, 36, 37, 35, 39, 34, 38, 37, 41, 36, 46, 37, 39, 42, 42, 41, 45, 40, 48, 44, 43, 42, 52, 45, 45, 46, 50, 45, 55, 48, 50, 49, 49, 50, 58, 49, 53, 54, 57

Offset: 1

Max Alekseyev, Dec 01 2003

nonn

Possibly related to Maillet's determinants.

			From _Alexander Adam_, Nov 10 2012: (Start)
a(2^m) = 2^(m-1) + m - 1.
Let p >= 3 be a prime number. Then a(p^m) = (p^m + 1) / 2 + m - 1.
a(625000) = a(2^3*5^7) = 2^2*5^7 + 4 * 8 - 2 = 312530. (End)

Alexander Adam, Proof of the formula
Wikipedia, Maillet's determinant

Cf. A218322, A055513, A203411, A000927.

a[n_] := MatrixRank[Table[Table[Mod[i * j, n], {j, 0, n - 1}], {i, 0, n - 1}]]; Array[a,100] (* Alexander Adam, Nov 10 2012 *)

PARI
```
a(n) = matrank(matrix(n,n,i,j,(i*j)%n))
```

Let n = Prod_{i>0} p_i^{m_i} be the prime factorization of n. Then a(n) = floor((n + 1)/2) + Prod_{i>0} (m_i + 1) - 2. - Alexander Adam, Nov 10 2012

a(n) = A000005(n) + A110654(n) - 2.

A035115 Relative class number h- of cyclotomic field Q(zeta_m) where m is n-th term of A035113.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 2, 3, 1, 8, 9, 5, 17, 8, 5, 9, 4, 37, 9, 7, 19, 19, 121, 10, 11, 55, 55, 11, 211, 43, 69, 201, 695, 64, 351, 13, 468, 39, 507, 156, 84, 75, 4889, 2593, 1536, 10752, 41241, 76301, 1280, 6795

Offset: 1

N. J. A. Sloane

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353.

Cf. A000010, A035113, A035114, A055513, A061494, A061653.

(* This is only a recomputation of the existing data. *)
terms = 76;
A035113 = Sort[Select[Range[3 terms], Mod[#, 4] != 2&], EulerPhi[#1] <= EulerPhi[#2]&];
A061653 = Import["https://oeis.org/A061653/b061653.txt", "Table"][[All, 2]];
a[n_] := A061653[[A035113[[n]] ]];
Array[a, terms] (* Jean-François Alcover, Aug 17 2019 *)

a(n) = A061653(A035113(n)).

Changed 41421 to 41241, value is given incorrectly in first edition of Washington, Matthew Johnson, Jul 20 2013

A061494 Relative class number h- of cyclotomic field Q(zeta_n) where n runs through positive integers not congruent to 2 (mod 4) [A042965, but omitting the initial 0].

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 8, 9, 1, 1, 1, 1, 37, 2, 1, 121, 211, 1, 1, 695, 1, 43, 5, 3, 4889, 10, 2, 9, 41241, 1, 76301, 7, 17, 64, 853513, 8, 69, 3882809, 3, 11957417, 11, 19, 1280, 100146415, 5, 2593, 838216959, 1, 6205, 1536, 55, 13379363737, 53872

Offset: 1

N. J. A. Sloane, Jun 16 2001

First edition of Washington incorrectly gives a(44) = h-(Q(zeta_59)) = 41421. [Matthew Johnson, Jul 20 2013]

			n=17: the 17th number not == 2 mod 4 is 23, and Q(zeta_23) = 3 is the first time that h- is bigger than 1, so a(17) = 3.

L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360.

Cf. A035115, A055513, A061653, A042965.

a(n) = A061653(A042965(n+1)). - M. F. Hasler, Feb 04 2009

Missing term a(1) = 1 inserted by N. J. A. Sloane, Feb 05 2009 at the suggestion of Tanya Khovanova and M. F. Hasler
More terms (from b-file of A061653), Joerg Arndt, Oct 07 2012
a(44) corrected by Matthew Johnson, Jul 20 2013

A230869 Primes p such that the class number h-tilde_p^{+} of the real cyclotomic field Q(zeta_p + zeta_p^(-1)) is greater than 1.

Original entry on oeis.org

163, 191, 229, 257, 277, 313, 349, 397, 401, 457, 491, 521, 547, 577, 607, 631, 641, 709, 733, 761, 821, 827, 829, 853, 857, 877, 937, 941, 953, 977, 1009, 1063, 1069, 1093, 1129, 1153, 1229, 1231, 1297, 1373, 1381, 1399, 1429, 1459, 1489, 1567, 1601, 1697, 1699, 1777, 1789, 1831, 1861, 1873, 1879, 1889, 1901, 1951

Offset: 1

N. J. A. Sloane, Nov 06 2013

nonn

Taken from the "Main Table" of Schoof.

There is a very slight chance that some primes are missing. In the unlikely event that the number that Schoof calls h-tilde_p is 1, while the actual class number h_p is actually not equal to 1, the prime p would be missing (see the Schoof and Miller articles for details).

John C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, arXiv:1405.1094 [math.NT], 2014.
John C. Miller, Real cyclotomic fields of prime conductor and their class numbers, arXiv:1407.2373 [math.NT], 2014.
Rene Schoof, Class numbers of real cyclotomic fields of prime conductor, Math. Comp., 72 (2002), 913-937.

Cf. A230870 (for the actual class numbers).

See also A000927, A055513, A035113, A035114, A035115.

A218322 Maillet determinant for prime(n).

Original entry on oeis.org

1, -5, 49, 14641, -371293, -410338673, 16983563041, 124279533640947, -82085029703668817512, 6812495416987166882889, -16890053810563300749953435929, -531714676529925182191868570093681, 98548851401030959947062957685234211, 4247541973383735863138308138153477847255, -62534081783371829558502906501683809565833328077, 1581923629964045589238110056212521488781448927448053161

Offset: 2

Max Alekseyev, Oct 25 2012

sign

Wikipedia, Maillet's determinant

Cf. A055513, A203411, A000927, A088922.

a(n) = p=prime(n); matdet(matrix((p-1)/2,(p-1)/2,i,j,(i/j)%p))

a(n) = (-p)^((p-3)/2) * h^-(p) = (-1)^((p-3)/2) * A203411(n) * A000927(n), where p=A000040(n).

cp's OEIS Frontend

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.

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A000927 "First factor" (or relative class number) h- for cyclotomic field Q( exp(2 Pi / prime(n)) ).

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A061653 Relative class number h- of cyclotomic field Q(zeta_n).

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A088922 Consider the n X n matrix with entries (i*j mod n), where i,j=0..n-1; a(n) = rank of this matrix over the real numbers.

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A035115 Relative class number h- of cyclotomic field Q(zeta_m) where m is n-th term of A035113.

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A061494 Relative class number h- of cyclotomic field Q(zeta_n) where n runs through positive integers not congruent to 2 (mod 4) [A042965, but omitting the initial 0].

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A230869 Primes p such that the class number h-tilde_p^{+} of the real cyclotomic field Q(zeta_p + zeta_p^(-1)) is greater than 1.

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A218322 Maillet determinant for prime(n).

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