A000367 -id:A000367 - 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

A300711 a(n) = A000367(n)/A001067(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 17, 13, 53, 1, 5, 7, 19, 29, 59, 1, 61, 31, 1, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37

Offset: 1

Bernd C. Kellner, Mar 11 2018

nonn

a(n) is the trivial factor of the numerator of Bernoulli(2n) that divides 2n.
The remaining part of the (unsigned) numerator equals a product of powers of irregular primes, or 1 if and only if n = 1, 2, 3, 4, 5, 7.
Alternatively, a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 does not divide 2n.

			a(5) = 5, since Bernoulli(10) = 5/66 and Bernoulli(10)/10 = 1/132.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

A111008 equals the first entries and slightly differs, see a(35).

Cf. A000367, A001067, A193267, A195989, A300330, A002445, A075180.

using Nemo
function A300711(n)
    b = bernoulli(n)
    div(numerator(b), numerator(b*QQ(1,n)))
end
[A300711(n) for n in 2:2:148] |> println # Peter Luschny, Mar 11 2018

A300711 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; P end: seq(A300711(n), n=1..74); # Peter Luschny, Mar 12 2018

Table[Numerator[BernoulliB[n]]/Numerator[BernoulliB[n]/n], {n, 2, 100, 2}]

a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ Jianing Song, Apr 05 2021

upto(N)=bernvec(N);forstep(n=2,2*N,2,print1(gcd(numerator(bernfrac(n)), n),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

a(n) = numerator(Bernoulli(2n))/numerator(Bernoulli(2n)/(2n)).
a(n) * A195989(n) = n. - Peter Luschny, Mar 12 2018
From Jianing Song, Apr 05 2021: (Start)
a(n) = gcd(numerator(Bernoulli(2n)), 2n).
a(n) = A002445(n)*(2n)/A075180(2n-1). (End)

A111008 a(n) = A000367(n)/A141590(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 629, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 1003, 481, 53, 1, 5, 7, 19, 29, 59, 1, 61, 2077, 103, 1, 65, 11, 67, 17, 23, 259, 71, 1, 73, 37, 25, 2489, 77

Offset: 0

Paul Curtz, Aug 25 2008

nonn

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..10000

See A141517.

Cf. A300711.

A120082 := proc(n) local b; if n = 0 then b := 1 ; elif n = 1 then b := -1/4 ; elif type(n,'odd') then b := 0; else b := bernoulli(n)/(n+1)! ; fi; numer(b) ; end:
A141590 := proc(n) A120082(2*n) ; end: A000367 := proc(n) numer(bernoulli(2*n)) ; end:
A111008 := proc(n) A000367(n)/A141590(n) ; end: seq(A111008(n),n=0..120) ; # R. J. Mathar, Sep 03 2009

upto(N)=bernvec(N);forstep(n=0,2*N,2,print1(gcd(numerator(bernfrac(n)), (n+1)!),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

Edited and extended by R. J. Mathar, Sep 03 2009

A035078 Numerators of partial sums of Bernoulli numbers B_{2n} = A000367/A002445.

Original entry on oeis.org

1, 7, 17, 81, 118, 2771, 4737, 63457, -1270924, 161636091, -464743285, 254905515589, -3006818262414, 299981806371451, -83955854172826681, 38482697321210434701, -1458143803622109300584, 83247435772128371635117

Offset: 0

N. J. A. Sloane

Index entries for sequences related to Bernoulli numbers.

A134825 Floor of the even-indexed Bernoulli numbers B_{2n} = A000367(n)/A002445(n).

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 1, -8, 54, -530, 6192, -86581, 1425517, -27298232, 601580873, -15116315768, 429614643061, -13711655205089, 488332318973593, -19296579341940069, 841693047573682615, -40338071854059455414, 2115074863808199160560, -120866265222965259346028

Offset: 0

Wolfdieter Lang, Nov 13 2007

			n=4: B_8=-1/30=-0,033... hence a(4)=-1.

C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall, 2006, p. 107.

Cf. A000367, A002445, A014509, A027641, A027642.

Floor@BernoulliB[2 Range[0, 20]] (* Vladimir Reshetnikov, Nov 12 2015 *)

vector(30, n, n--; floor(bernfrac(2*n))) \\ Altug Alkan, Nov 12 2015

a(n) = floor(B_(2n)), n>=0, with B_{2n} = A000367(n)/A002445(n) = A027641(2n)/A027642(2n).

A176546 Bernoulli numerators A000367 with an additional 1 inserted to represent B_1.

Original entry on oeis.org

1, 1, 1, -1, 1, -1, 5, -691, 7, -3617, 43867, -174611, 854513, -236364091, 8553103, -23749461029, 8615841276005, -7709321041217, 2577687858367, -26315271553053477373, 2929993913841559, -261082718496449122051

Offset: 0

Paul Curtz, Apr 20 2010

Equivalent to adding a 1 in front of A000367, or removing zeros in A164555.
(One could also remove zeros in A027641 which would flip the sign of a(1)).
The denominators are in A006954.

			B_0=1/1, B_1=1/2 "originally", B_2=1/6, B_4=-1/30, B_6=1/42,...

Cf. A165908, A174276.

A180943 Odd composite numbers m for which 12|A000367((m+1)/2)|==(-1)^{(m-1)/ 2} A002445((m+1)/2) (mod m).

Original entry on oeis.org

33, 169, 481, 561, 793, 805, 949, 1105, 1261, 1417, 1645, 1729, 2041, 2353, 2465, 2509, 2821, 2977, 3133, 3421, 3445, 3601, 4069, 4123, 4381, 4537, 4849, 5161, 5317, 5473, 5629, 5941, 6061, 6205, 6601, 7033, 7093, 7189, 7501, 7813, 7885, 7969, 8113

Offset: 1

Vladimir Shevelev, Sep 27 2010

nonn

These are pseudoprimes in the sense that the congruence of the definition is valid if any odd prime is substituted for m.
Entries of the form m = 4*k+3 are apparently rare: 4123, 8911, ...
Computed to 50 terms by D. S. McNeil, Sep 05 2010.

V. Shevelev, B-pseudoprimes, seqfan list, Sep 04 2010
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. [_N. J. A. Sloane_, Feb 07 2013]

Cf. A000367, A002445, A180942.

A000367 := proc(n) numer(bernoulli(2*n)) ; end proc:
A002445 := proc(n) denom(bernoulli(2*n)) ; end proc:
isA180943 := proc(m) if type(m,'odd') and not isprime(m) then 12*abs(A000367((m+1)/2)) mod m = (-1)^((m-1)/2)*A002445((m+1)/2) mod m ; else false; end if; end proc:
A180943 := proc(n) option remember; if n = 1 then 33; else for a from procname(n-1)+2 by 2 do if isA180943(a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Oct 24 2010

nb[n_] := Numerator[BernoulliB[2n]];
db[n_] := Denominator[BernoulliB[2n]];
okQ[m_] := CompositeQ[m] && Mod[12*Abs[nb[(m+1)/2]], m] == Mod[(-1)^((m-1)/2)*db[(m+1)/2], m];
Select[Range[33, 9999, 2], okQ] (* Jean-François Alcover, Feb 28 2024 *)

Comments rephrased and program added by R. J. Mathar, Oct 24 2010

Typo in data fixed by Jean-François Alcover, Feb 28 2024

A035077 Denominators of partial sums of Bernoulli numbers B_{2n} = A000367/A002445.

Original entry on oeis.org

1, 6, 15, 70, 105, 2310, 5005, 30030, 255255, 3233230, 969969, 44618574, 37182145, 223092870, 3234846615, 66853496710, 100280245065, 200560490130, 1236789689135, 7420738134810, 30425026352721

Offset: 0

N. J. A. Sloane

Index entries for sequences related to Bernoulli numbers.

Cf. A035078.

Denominator[Accumulate[BernoulliB[2*Range[0,20]]]] (* Harvey P. Dale, May 24 2015 *)

A081739 a(n) = (-1/2^n)*(A000367(2^n)+1).

Original entry on oeis.org

0, 0, 452, 481832565076, 3336994692120829058949545155598207748188, 4183667308477313795108662587270037418426429056133620947343416969739965511482242658521291610836247844507748569589344

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

Cf. A000367, A076547.

a[n_] := -(Numerator[BernoulliB[2^(n+1)]] + 1)/2^n; Array[a, 6] (* Amiram Eldar, May 03 2025 *)

a(n) = -(numerator(bernfrac(1 << (n+1))) + 1) >> n; \\ Amiram Eldar, May 03 2025

a(n) = A076547(n)-1 (Conjecture). - R. J. Mathar, Apr 22 2007

A308765 Irregular triangle T(n,k) read by rows with 1 <= k <= A091887 even indices 2i such that n-th irregular prime p (A000928) divides the numerator of the Bernoulli numbers B_{2i} (A000367) with 0 <= 2i <= p-3.

Original entry on oeis.org

32, 44, 58, 68, 24, 22, 130, 62, 110, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 300, 100, 174, 200, 382, 126, 240, 366, 196, 130, 94, 194, 292, 336, 338, 400, 86, 270, 486, 222, 52, 90, 92, 22, 592, 522, 20, 174, 338, 428, 80, 226, 236, 242, 554, 48, 224, 408, 502, 628, 32, 12, 200, 378, 290, 514, 260, 732, 220, 330, 628, 544, 744, 102, 66, 868, 162, 418, 520, 820, 156, 166

Offset: 1

Martin Renner, Jun 23 2019

First index T(n,1) in row n is A035112(n).

			Triangle starts with
n = 1 => p = 37 divides the numerator of B_{32} = -7709321041217;
n = 2 => p = 59: B_{44};
n = 3 => p = 67: B_{58};
n = 4 => p = 101: B_{68};
n = 5 => p = 103: B_{24};
n = 6 => p = 131: B_{22};
n = 7 => p = 149: B_{130};
n = 8 => p = 157: B_{62}, B_{110};
n = 9 => p = 233: B_{84};
etc.

Cf. A000367, A000928, A035112, A060974, A060975, A073276, A073277, A091887.

T:=[]:
for j from 2 to 168 do
  p:=ithprime(j);
  B:=[]:
  for i from 1 to (p-3)/2 do
    if type(numer(bernoulli(2*i))/p,integer) then B:=[op(B),2*i]: fi:
  od:
  T:=[op(T),op(B)];
od:
op(T);

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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A300711 a(n) = A000367(n)/A001067(n).

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A111008 a(n) = A000367(n)/A141590(n).

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A035078 Numerators of partial sums of Bernoulli numbers B_{2n} = A000367/A002445.

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A134825 Floor of the even-indexed Bernoulli numbers B_{2n} = A000367(n)/A002445(n).

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A176546 Bernoulli numerators A000367 with an additional 1 inserted to represent B_1.

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A180943 Odd composite numbers m for which 12*|A000367((m+1)/2)|==(-1)^{(m-1)/ 2}* A002445((m+1)/2) (mod m).

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A035077 Denominators of partial sums of Bernoulli numbers B_{2n} = A000367/A002445.

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A180943 Odd composite numbers m for which 12|A000367((m+1)/2)|==(-1)^{(m-1)/ 2} A002445((m+1)/2) (mod m).