A090495 -id:A090495 - cp's OEIS frontend

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

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

A046968 Numerators of coefficients in Stirling's expansion for log(Gamma(z)).

Original entry on oeis.org

Offset: 1

Douglas Stoll (dougstoll(AT)email.msn.com)

A001067(n) = a(n) if n<574; A001067(574) = 37*a(574). - Michael Somos, Feb 01 2004

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205

Seiichi Manyama, Table of n, a(n) for n = 1..314
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. 257, Eq. 6.1.41.
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
Eric Weisstein's World of Mathematics, Stirling's Series
Index entries for sequences related to Bernoulli numbers.

Denominators given by A046969.

Similar to but different from A001067. See A090495, A090496.

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

[Numerator(Bernoulli(2*n)/(2*n*(2*n-1))): n in [1..25]]; // G. C. Greubel, Sep 19 2019

seq(numer(bernoulli(2*n)/(2*n*(2*n-1))), n = 1..25); # G. C. Greubel, Sep 19 2019

Table[ Numerator[ BernoulliB[2n]/(2n(2n - 1))], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *)
s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^42; DeleteCases[CoefficientList[s, 1/z], 0] // Numerator (* Jean-François Alcover, Jun 13 2017 *)

a(n)=if(n<1,0,numerator(bernfrac(2*n)/(2*n)/(2*n-1)))

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

From numerator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2*Pi)*z^(z-0.5)*exp(-z)*exp(J(z)).

More terms from Frank Ellermann, Jun 13 2001

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

A092291 Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2m)/(2m)) / numerator(Bernoulli(2m)/(2m(2m-1))) equals p.

Original entry on oeis.org

574, 1269, 1910, 3384, 1185, 1376, 9611, 4789, 9670, 20946, 13019, 11247, 2689, 22708, 13355, 45251, 48407, 32653, 18761, 38706, 76391, 25563, 50310, 79023, 44948, 29864, 21716, 71441, 104339, 22993, 73572, 61549, 14714, 26122, 6227, 179369, 159687, 5862, 132157, 24925, 76023, 15346, 73479, 136956, 212240, 10587, 3801, 137040, 108520, 194171, 98550, 282532, 87272, 133081, 220187, 305002, 41764, 27268, 380180, 70921, 184940, 241076, 73858, 80108, 250927

Offset: 1

N. J. A. Sloane, based on a suggestion of Roland Bacher, Feb 05 2004

It was conjectured that a(n) = (1 + A000928(n) * (A035112(n) - 1))/2. However, Bernd Kellner's insightful paper shows that this formula first fails for the irregular prime 6449. - T. D. Noe, Feb 10 2004

Bernd Kellner, A conjecture about numerators of Bernoulli numbers

Term in A090495 corresponding to first occurrence of p in A090496.

(* This program is not convenient for a large number of terms *) irregularPrimeQ[p_] := Module[{k = 1}, While[2*k <= p-3 && Mod[ Numerator[ BernoulliB[2*k]], p] != 0, k++]; 2*k <= p-3]; irregularPrime[1] = 37; irregularPrime[n_] := irregularPrime[n] = Module[{p}, For[p = NextPrime[ irregularPrime[n-1]], True, p = NextPrime[p], If[ irregularPrimeQ[p], Return[p]]]]; a[n_] := a[n] = For[m = 1, True, m++, If[ Numerator[BernoulliB[2*m]/(2*m)] / Numerator[ BernoulliB[2*m]/(2*m*(2*m-1))] == irregularPrime[n], Return[m]]]; Table[ Print[a[n]]; a[n], {n, 1, 15}] (* Jean-François Alcover, Sep 27 2013 *)

Initial terms were computed by Roland Bacher, Feb 04 2004; further terms from Hans Havermann, Feb 05 2004 and T. D. Noe, Feb 06 2004

Offset modified by Jean-François Alcover, Sep 27 2013

A090177 Numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n+1))).

Original entry on oeis.org

610, 1276, 1287, 1327, 1506, 1942, 1976, 2608, 2971, 3038, 3274, 3484, 3940, 4187, 4491, 4606, 4749, 4945, 5272, 5938, 6398, 6460, 6478, 6540, 6604, 6819, 7270, 7936, 8171, 8534, 8602, 8609, 8713, 9268, 9759, 9882, 9902, 9934, 10021

Offset: 1

mohammed bouayoun (bouyao(AT)wanadoo.fr), Feb 04 2004

nonn

Coincides with A090495(n) + A090496(n) - 1, except for order of terms.

Cf. A090179, A001067, A090495.

Do[ If[ Numerator[ BernoulliB[2n] / (2n)] != Numerator[ BernoulliB[2n] / (2n(2n + 1))], Print[n]], {n, 1, 10030}]

a(6)-a(39) from Robert G. Wilson v, Feb 09 2004

A090179 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, 37, 103, 59, 131, 37, 67, 37, 283, 59, 37, 101, 37, 67, 691, 37, 59, 157, 37, 37, 67, 59, 617, 103, 37, 593, 37, 37, 59, 101, 37, 67, 157, 37, 149, 59, 233, 37, 131

Offset: 1

mohammed bouayoun (bouyao(AT)wanadoo.fr), Feb 04 2004

nonn

Cf. A090177, A001067, A090495.

Do[ If[ r = Numerator[ BernoulliB[2n]/(2n)] / Numerator[ BernoulliB[2n]/(2n(2n + 1))]; r != 1, Print[r]], {n, 1, 10030}]

a(6)-a(39) from Robert G. Wilson v, Feb 09 2004

A090790 Numbers r arising in A090791.

Original entry on oeis.org

30, 42, 56, 66, 22, 20, 128, 60, 82, 162, 98, 82, 18

Offset: 1

Cino Hilliard, Feb 16 2004

These values of r correspond to the first 13 irregular primes produced by a/b.

			Given a, b as defined above and p=37, r=30, 52 = pk+r/2 = 37*1 + 30/2 is the smallest number that for a<>b a/b = 37.

Cf. A090495, A090496, A090791.

bern3(m,r) = { for(i=m,m, p=irprime(i); /* use the Somos script below to get irregular prime */ for(k=1,p, if(r%2,n=p*k+(p+r)/2,n=p*k+r/2); n2=n+n; a = numerator(bernfrac(n2)/(n2));
b = numerator(a/(n2-r)); v=a/b; if(a <> b && v==p,print(k","n","v);break) ) ) } /* A001067 */

irprime(n) = { my(p); if(n<1, 0, p=irprime(n-1) + (n==1); while(p = nextprime(p+2), forstep(i=2, p-3, 2, if( numerator(bernfrac(i))%p == 0, break(2)))); p) }; /* compute irregular primes irprime from - Michael Somos, Feb 04 2004 */

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p*k+(p+r)/2 if r is odd and n = p*k+r/2 if r is even where k = 1, 2.. For every irregular prime p there is an r such that n is minimum.

A090791 Minimal numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n-r))) for some integer r.

Original entry on oeis.org

52, 80, 95, 134, 114, 141, 213, 187, 274, 338, 312, 312, 292

Offset: 1

Cino Hilliard, Feb 16 2004

These values of n correspond to the first 13 irregular primes produced by a/b.

			Given a,b as defined above and p=37,r=30, n=pk+r/2 = 37*k + 30/2 = 37k+15 = 52 = the smallest number that for a<>b a/b = 37.

Cf. A090790, A090495, A090496.

bern3(m,r) = { for(i=m,m, p=irprime(i); /* use the Somos script below to get irregular prime */ for(k=1,p, if(r%2,n=p*k+(p+r)/2,n=p*k+r/2); n2=n+n; a = numerator(bernfrac(n2)/(n2));
b = numerator(a/(n2-r)); v=a/b; if(a <> b && v==p,print(k","n","v);break) ) ) } /* A001067 */

irprime(n) = { my(p); if(n<1, 0, p = irprime(n-1) + (n==1); while(p = nextprime(p+2), forstep(i=2, p-3, 2, if( numerator(bernfrac(i))%p == 0, break(2)))); p) };  /* compute irregular primes irprime from - Michael Somos, Feb 04 2004 */

A090798 Irregular primes in the ratio numerator(Bernoulli(2n)/(2n)) / numerator(Bernoulli(2n)/(2n(2n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1.

Original entry on oeis.org

37, 59, 67, 101, 103, 131, 149, 157, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 353, 379, 379, 389, 401, 409, 421, 433, 461, 463, 467, 467, 491, 491, 491, 523, 541, 547, 547, 557, 577, 587, 587, 593, 607, 613, 617, 617, 617, 619, 631, 631, 647

Offset: 1

Cino Hilliard, Feb 16 2004

nonn

Only even values of r need to be tested.

See Table A.3, "Calculated irregular pairs of order 10 of primes below 1000," in B. C. Kellner.

Robert G. Wilson v, Table of n, a(n) for n = 1..2000
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

Cf. A090495 A090496.

f[p_] := Block[{c = 0, k = 1}, While[ 2k <= p - 3, If[ Mod[ Numerator@ BernoulliB[ 2k], p] == 0, c++]; k++]; c]; p = 5; lst = {}; While[p < 1001, AppendTo[lst, Table[p, {f@ p}]]; p = NextPrime@ p]; Flatten@ lst

\ prestore some ireg primes in iprime[] bernmin(m) = { for(x=1,m, p=iprime[x]; forstep(r=2,p,2, n=r/2+p; n2=n+n; a = numerator(bernfrac(n2)/(n2)); \ A001067 b = numerator(a/(n2-r)); \ if(a <> b,print(r","n","a/b)) if(a <> b,print1(a/b",")) ) ) }

Given a = numerator(Bernoulli(2*n)/(2*n)) and b = numerator(a/(2*n-r)) for integer r positive or negative, then n>0 n = p + r/2 For every irregular prime p there is an r such that n is minimum.

cp's OEIS Frontend

A274297 a(n) = 2*A090495(n) - 1.

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

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A046968 Numerators of coefficients in Stirling's expansion for log(Gamma(z)).

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

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A092291 Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2*m)/(2*m)) / numerator(Bernoulli(2*m)/(2*m*(2*m-1))) equals p.

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A090177 Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n+1))).

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A090179 Ratio of numerator(Bernoulli(2*n)/(2*n)) to numerator(Bernoulli(2*n)/(2*n*(2*n+1))) for n's for which they are different.

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A090790 Numbers r arising in A090791.

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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.

A092291 Let p = n-th irregular prime, A000928(n). Then a(n) = smallest value of m such that numerator(Bernoulli(2m)/(2m)) / numerator(Bernoulli(2m)/(2m(2m-1))) equals p.

A090177 Numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n+1))).

A090179 Ratio of numerator(Bernoulli(2n)/(2n)) to numerator(Bernoulli(2n)/(2n(2n+1))) for n's for which they are different.

A090791 Minimal numbers n such that numerator(Bernoulli(2n)/(2n)) is different from numerator(Bernoulli(2n)/(2n(2n-r))) for some integer r.

A090798 Irregular primes in the ratio numerator(Bernoulli(2n)/(2n)) / numerator(Bernoulli(2n)/(2n(2n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1.