A000928 -id:A000928 - cp's OEIS frontend

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.

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);

A001067 Numerator of Bernoulli(2n)/(2n).

Original entry on oeis.org

1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691, -2530297234481911294093

Offset: 1

N. J. A. Sloane, Richard E. Borcherds (reb(AT)math.berkeley.edu)

It was incorrectly claimed that a(n) is "also numerator of "modified Bernoulli number" b(2n) = Bernoulli(2*n)/(2*n*n!)"; actually, the numerators of these fractions and the numerators of "modified Bernoulli numbers" (see A057868 for details) differ from each other and from this sequence. - Andrey Zabolotskiy, Dec 03 2022
Ramanujan incorrectly conjectured that the sequence contains only primes (and 1). - Jud McCranie. See A112548, A119766.
a(n) = A046968(n) if n < 574; a(574) = 37 * A046968(574). - Michael Somos, Feb 01 2004
Absolute values give denominators of constant terms of Fourier series of meromorphic modular forms E_k/Delta, where E_k is the normalized k th Eisenstein series [cf. Gunning or Serre references] and Delta is the normalized unique weight-twelve cusp form for the full modular group (the generating function of Ramanujan's tau function.) - Barry Brent (barrybrent(AT)iphouse.com), Jun 01 2009
|a(n)| is a product of powers of irregular primes (A000928), with the exception of n = 1,2,3,4,5,7. - Peter Luschny, Jul 28 2009
Conjecture: If there is a prime p such that 2*n+1 < p and p divides a(n), then p^2 does not divide a(n). This conjecture is true for p < 12 million. - Seiichi Manyama, Jan 21 2017

			The sequence Bernoulli(2*n)/(2*n) (n >= 1) begins 1/12, -1/120, 1/252, -1/240, 1/132, -691/32760, 1/12, -3617/8160, ...
The sequence of modified Bernoulli numbers begins 1/48, -1/5760, 1/362880, -1/19353600, 1/958003200, -691/31384184832000, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 259, (6.3.18) and (6.3.19); also p. 810.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
R. Kanigel, The Man Who Knew Infinity, pp. 91-92.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 285.
J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, p. 93.

Seiichi Manyama, Table of n, a(n) for n = 1..314 (first 100 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 259, (6.3.18) and (6.3.19).
D. Bar-Natan, T. T. Q. Le and D. P. Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, arXiv:math/0204311 [math.QA], 2002-2003; Geometry and Topology 7-1 (2003) 1-31.
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.
E. Z. Goren, Tables of values of Riemann zeta functions
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Bernoulli Number.
Wikipedia, Kummer-Vandiver conjecture
Index entries for sequences related to Bernoulli numbers

Similar to but different from A046968. See A090495, A090496.
Denominators given by A006953.
Cf. A000367, A006863, A033563, A046968.
Cf. A141590, A255505.

List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n)));  # G. C. Greubel, Sep 19 2019

[Numerator(Bernoulli(2*n)/(2*n)):n in [1..40]]; // Vincenzo Librandi, Sep 17 2015

A001067_list := proc(n) 1/(1-1/exp(z)); series(%,z,2*n+4);
seq(numer((2*i+1)!*coeff(%,z,2*i+1)),i=0..n) end:
A001067_list(21); # Peter Luschny, Jul 12 2012

Table[ Numerator[ BernoulliB[2n]/(2n)], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *)

{a(n) = if( n<1, 0, numerator( bernfrac(2*n) / (2*n)))}; /* Michael Somos, Feb 01 2004 */

@CachedFunction
def S(n, k) :
    if k == 0 :
        if n == 0 : return 1
        else: return 0
    return S(n, k-1) + S(n-1, n-k)
def BernoulliDivN(n) :
    if n == 0 : return 1
    return (-1)^n*S(2*n-1,2*n-1)/(4^n-16^n)
[BernoulliDivN(n).numerator() for n in (1..22)]
# Peter Luschny, Jul 08 2012

[numerator(bernoulli(2*n)/(2*n)) for n in (1..25)] # G. C. Greubel, Sep 19 2019

Zeta(1-2*n) = - Bernoulli(2*n)/(2*n).
G.f.: numerators of coefficients of z^(2*n) in z/(exp(z)-1). - Benoit Cloitre, Jun 02 2003
For 2 <= k <= 1000 and k != 7, the 2-order of the full constant term of E_k/Delta = 3 + ord_2(k - 7). - Barry Brent (barrybrent(AT)iphouse.com), Jun 01 2009
G.f. for Bernoulli(2*n)/(2*n) = a(n)/A006953(n): (-1)^n/((2*Pi)^(2*n)*(2*n))*integral(log(1-1/t)^(2*n) dt,t=0,1). - Gerry Martens, May 18 2011
E.g.f.: a(n) = numerator((2*n+1)!*[x^(2*n+1)](1/(1-1/exp(x)))). - Peter Luschny, Jul 12 2012
|a(n)| = numerator of Integral_{r=0..1} HurwitzZeta(1-n, r)^2 dr. More general: |Bernoulli(2*n)| = binomial(2*n,n)*n^2*I(n) for n >= 1 where I(n) denotes the integral. - Peter Luschny, May 24 2015

A091216 Numbers k such that numerator of Bernoulli(2*k) is divisible by 37, the first irregular prime.

Original entry on oeis.org

16, 34, 37, 52, 70, 74, 88, 106, 111, 124, 142, 148, 160, 178, 185, 196, 214, 222, 232, 250, 259, 268, 286, 296, 304, 322, 333, 340, 358, 370, 376, 394, 407, 412, 430, 444, 448, 466, 481, 484, 502, 518, 520, 538, 555, 556, 574, 592, 610, 628

Offset: 1

N. J. A. Sloane, Feb 24 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Seiichi Manyama)

Cf. A000928, A092221, A092222, A092223, A092224, A092225, A092226, A092227, A092228, A092229, A092230, A092231.

Select[ Range[ 681], Mod[ Numerator[ BernoulliB[2# ]], 37] == 0 &] (* Robert G. Wilson v, Feb 24 2004 *)

for(j=1,260, if (! (numerator(bernfrac(2*j))%37), print1(j, ", ")))

More terms from Robert G. Wilson v, Feb 24 2004

A073277 Irregular primes with irregularity index two.

Original entry on oeis.org

157, 353, 379, 467, 547, 587, 631, 673, 691, 809, 929, 1291, 1297, 1307, 1663, 1669, 1733, 1789, 1933, 1997, 2003, 2087, 2273, 2309, 2371, 2383, 2423, 2441, 2591, 2671, 2789, 2909, 2957, 3391, 3407, 3511, 3517, 3533, 3539, 3559, 3593, 3617, 3637, 3851

Offset: 1

Robert G. Wilson v, Jul 22 2002

nonn

Subsequence of A060974.

T. D. Noe, Table of n, a(n) for n=1..10000 (from Buhler et al.)
J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. A. Shokrollahi, Irregular Primes and Cyclotomic Invariants to 12 Million, J. Symbolic Computation 31, 2001, 89-96.
Bernoulli numbers, irregularity index of primes

Cf. A000928, A091887, A000367, A060974, A060975 and A073276.

Do[p = Prime[n]; k = 1; c = 0; While[ 2*k < p - 3, If[ Mod[ Numerator[ BernoulliB[2*k]], p] == 0, c++ ]; k++ ]; If[ c == 2, Print[p]], {n, 3, 550} ]

A092221 Numbers k such that numerator of Bernoulli(2*k) is divisible by 59, the second irregular prime.

Original entry on oeis.org

22, 51, 59, 80, 109, 118, 138, 167, 177, 196, 225, 236, 254, 283, 295, 312, 341, 354, 370, 399, 413, 428, 457, 472, 486, 515, 531, 544, 573, 590, 602, 631, 649, 660, 689, 708, 718, 747, 767, 776, 805, 826, 834, 863, 885, 892, 921, 944, 950, 979, 1003, 1008

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Satisfies a(n) = 2*a(n-3) - a(n-6) for n < 67. - Chai Wah Wu, May 28 2016

Amiram Eldar, Table of n, a(n) for n = 1..6450 (terms 1..1000 from Joerg Arndt)
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092222, A092223, A092224, A092225, A092226, A092227, A092228, A092229, A092231.

Select[ Range[ 1036], Mod[ Numerator[ BernoulliB[2# ]], 59] == 0 &]

for(n=0, 10^3, if( numerator(bernfrac(2*n))%59==0, print1(n, ", ") ) ); \\ Joerg Arndt, May 29 2016

from sympy import bernoulli
A092221_list = [n for n in range(10**3) if not bernoulli(2*n).p % 59] # Chai Wah Wu, May 28 2016

A060975 Irregular primes with irregularity index three.

Original entry on oeis.org

491, 617, 647, 1151, 1217, 1811, 1847, 2939, 3833, 4003, 4657, 4951, 6763, 7687, 8831, 9011, 10463, 10589, 12073, 13217, 14533, 14737, 14957, 15287, 15787, 15823, 16007, 17681, 17863, 18713, 18869, 20533, 20939, 24019, 24659, 25153, 26561

Offset: 1

Robert G. Wilson v, Jul 22 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated by Hart et al., terms 1..9824 from T. D. Noe, calculated by Buhler et al.)
J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. A. Shokrollahi, Irregular Primes and Cyclotomic Invariants to 12 Million, J. Symbolic Computation 31, 2001, 89-96.
Bernoulli numbers, irregularity index of primes
William Hart, David Harvey and Wilson Ong, Irregular primes to two billion, Mathematics of Computation, Vol. 86, No. 308 (2017), pp. 3031-3049; also available at arXiv:1605.02398 [math.NT], 2016.
David Harvey, Irregular primes to two billion (includes a list of all primes less than 2^31).

Cf. A000928, A000367, A060974, A060975, A073276 and A073277.

Do[p = Prime[n]; k = 1; c = 0; While[ 2*k < p - 3, If[ Mod[ Numerator[ BernoulliB[2*k]], p] == 0, c++ ]; k++ ]; If[ c == 3, Print[p]], {n, 3, 1000} ]
Do[p = Prime@n; k = 1; c = 0; While[ 2*k < p - 3, If[ Mod[ Numerator[ BernoulliB[2*k]], p] == 0, c++ ]; k++ ]; If[ c == 3, Print@p], {n, 3, 13887} ]

Extended by Robert G. Wilson v, Sep 20 2006

A007703 Regular primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431

Offset: 1

N. J. A. Sloane, Simon Plouffe

A prime p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 223-227.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary, Regular prime.
K. Conrad, Fermat's Last Theorem For Regular Primes.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
F. Luca, A. Pizarro-Madariaga, and C. Pomerance, On the counting function of irregular primes, 2014.
O. A. Ivanova, Regular prime number.
D. Jao, PlanetMath.org, Regular prime.
A. L. Robledo, PlanetMath.org, Examples of regular primes.
Eric Weisstein's World of Mathematics, Regular Prime.
Bernoulli numbers, irregularity index of primes

Cf. A000928 (irregular primes) and A061576 for further references.

s = {}; Do[p = Prime@n; k = 1; While[2k <= p - 3 && Mod[Numerator@BernoulliB[2k], p] != 0, k++ ]; If[2k > p - 3, AppendTo[s, p]], {n, 2, 80}]; s (* Robert G. Wilson v Sep 20 2006 *)

is(p)=forstep(k=2,p-3,2,if(numerator(bernfrac(k))%p==0, return(0))); isprime(p) \\ Charles R Greathouse IV, Feb 25 2014

from sympy import prime, isprime, bernoulli
def ok(n):
    for k in range(2, n - 2, 2):
        if bernoulli(k).as_numer_denom()[0] % n == 0:
            return 0
    return isprime(n)
[n for n in range(3, 501) if ok(n)] # Indranil Ghosh, Jun 27 2017, after Charles R Greathouse IV

Corrected by Gerard Schildberger, Jun 01 2004

A090496 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n-1))) for n's for which they are different.

Original entry on oeis.org

37, 103, 37, 59, 131, 37, 67, 37, 283, 59, 37, 101, 691, 37, 67, 37, 59, 157, 37, 617, 37, 593, 67, 59, 103, 37, 37, 37, 59, 101, 67, 157, 37, 37, 149, 233, 59, 131, 37, 37, 683, 67, 37, 271, 59, 103, 37, 37, 67, 263, 37, 59, 307, 101, 37, 37, 577, 59, 67, 37, 653, 37, 37, 59, 103, 157, 37, 67, 37, 59, 131, 101

Offset: 1

N. J. A. Sloane, Feb 03 2004

A001067(n) / A046968(n) when they are different, or alternatively, gcd(A001067(n),2n-1) when that number is > 1.
These numbers are always products of irregular primes (A000928).
All values yielding 37 are of the form 574+666*k, k=0,1,2,3,4,... and form thus an arithmetic progression with step 666=18*37=((37-1)/2)*37. All values yielding 59 are of the form 1269+1711*k, k=0,1,2,3 and 1711=28*59=((59-1)/2)*59. The two values yielding 67 are at distance 2211=((67-1)/2)*67. Conjecture: all indices yielding a given prime p form an arithmetic progression of step ((p-1)/2)*p. See A092291. - Roland Bacher, Feb 04 2004
The positions where 37 occurs appear to coincide with A026352. - Mohammed Bouayoun, Feb 05 2004
Roland Bacher conjectures that values of n yielding the same quotient p form an arithmetic progression n0+d*k, where d = p(p-1)/2. Actual and conjectured values of n0 are in the sequence A092291.
Composite values do occur. An example is 2n = 272876, which yields a quotient of 37*59. This was found by tdn using the Kummer congruences and CRT: using the irregular pairs (37,32) and (59,44), we know that the following Diophantine equations must be solved for (k,l,m): 32+36*k = 44+58*l = 1+37*59*m. Some quotients are not possible, e.g., 37*67, 37*103. All quotients are the product of irregular primes A000928. Composite quotients imply there are missing terms in the arithmetic progression conjectured by Bacher. - T. D. Noe, Feb 12 2004

Amiram Eldar, Table of n, a(n) for n = 1..200 (calculated from the b-file at A090495)
Bernd Kellner, A conjecture about numerators of Bernoulli numbers
Eric Weisstein's World of Mathematics, Stirling's Series

Cf. A090495, A001067, A046968, A092291.

A090496 = {}; Do[ r = Numerator[ b = BernoulliB[2n]/(2n) ] / Numerator[ b/(2n-1) ]; If[ r > 1, Print[n, " ", r]; AppendTo[ A090496, r] ], {n, 1, 20000}]; A090496 (* Jean-François Alcover, Jan 24 2012 *)

a(1)-a(7) from Michael Somos and W. Edwin Clark, Feb 03 2004
a(8), a(9) from Robert G. Wilson v, Feb 03 2004
a(10)-a(12) from Eric W. Weisstein, Feb 03 2004
a(13)-a(39) from Cino Hilliard, Feb 03 2004
a(40)-a(44) from Eric W. Weisstein, Feb 04 2004
Terms from a(45) onwards from David Wasserman, Dec 06 2005

cp's OEIS Frontend

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.

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A001067 Numerator of Bernoulli(2*n)/(2*n).

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A091216 Numbers k such that numerator of Bernoulli(2*k) is divisible by 37, the first irregular prime.

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A073277 Irregular primes with irregularity index two.

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A092221 Numbers k such that numerator of Bernoulli(2*k) is divisible by 59, the second irregular prime.

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A060975 Irregular primes with irregularity index three.

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A007703 Regular primes.

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A001067 Numerator of Bernoulli(2n)/(2n).

A090496 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n-1))) for n's for which they are different.