A035112 -id:A035112 - cp's OEIS frontend

A189683 Irregular pairs (p,2k) ordered by increasing k.

691, 12, 3617, 16, 43867, 18, 283, 20, 617, 20, 131, 22, 593, 22, 103, 24, 2294797, 24, 657931, 26, 9349, 28, 362903, 28, 1721, 30, 1001259881, 30, 37, 32, 683, 32, 305065927, 32, 151628697551, 34, 26315271553053477373, 36, 154210205991661, 38, 137616929, 40

Offset: 1

Jonathan Sondow, Apr 25 2011

nonn

The subsequence of irregular primes p is A046753.

			The first few irregular pairs are (691,12), (3617,16), (43867,18), (283,20), (617,20), (131,22), (593,22), ...

T. D. Noe, Table of n, a(n) for n = 1..132
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.
B. Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Amer. Math. Soc., 48 (2011), 155-209. See footnote 64 on p. 205.
Wikipedia, Regular prime: Irregular pairs.

Cf. A000367, A000928, A035112, A046753, A189684, A189685.

Flatten[Table[p = Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[n]]]], # >= n+3 &]; Transpose[{p, Table[n, {Length[p]}]}], {n, 2, 70, 2}]] (* T. D. Noe, Apr 25 2011 *)

A092681 Least number 2k such that p^2 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

Original entry on oeis.org

284, 914, 3292, 5768, 228, 12112, 11082, 6302, 11498, 40220, 34724, 51976, 1434, 74750, 67316, 21508, 63532, 39378, 67066, 64012, 91576, 137766, 137552, 105582, 158838, 147660, 175758, 194776, 173842, 102148, 132072, 107112, 127736, 29248

Offset: 1

T. D. Noe, Mar 03 2004

nonn

The ordered-pair (p(n),a(n)), where p(n) is the n-th irregular prime, is called a second-order irregular pair. Some irregular primes, such as 157, have more than one pair. See A091887 for a count of the pairs for each irregular prime.

Bernd Kellner, On irregular pairs of higher order (in German)

Cf. A000928, A035112, A091887, A092682.

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

A189684 Number of irregular pairs (p,2n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 1, 1, 2, 1, 4, 2, 3, 2, 4, 3, 3, 4, 3, 4, 2, 2, 4, 3, 2, 1, 2, 3, 5, 3, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 3, 4, 3, 5, 3, 1, 4, 1, 3, 3, 7, 3, 6, 5, 3, 4, 3, 7, 6, 3, 2, 6, 6, 6, 5, 6, 5, 7, 5, 4, 7, 8, 5, 3, 2, 7

Offset: 1

Jonathan Sondow, Apr 25 2011

nonn

a(n) is the number of primes p >= 2n+3 that divide the numerator of the Bernoulli number B_{2n}.

B. Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Amer. Math. Soc., 48 (2011), 155-209. See footnote 64 on p. 205.
Samual S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers
Wikipedia, Irregular pairs

Cf. A000367, A000928, A035112, A046753, A189683, A189685.

Table[p = Select[First /@ FactorInteger[Abs[Numerator[BernoulliB[2n]]]], # >= 2*n+3 &]; Length[p], {n, 35}] (* T. D. Noe, Apr 25 2011 *)

A092682 Least number 2k such that p^3 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

Original entry on oeis.org

37580, 86464, 153640, 581468, 914250, 454892, 1510618, 3557642, 84974, 8905404, 11482532, 9629910, 1025814, 9252440, 6484016, 22003936, 17706562, 30054878, 18332698, 37340812, 39775150, 31082358, 5118308, 20315982, 57395934, 25079280

Offset: 1

T. D. Noe, Mar 03 2004

nonn

The ordered-pair (p(n),a(n)), where p(n) is the n-th irregular prime, is called a third-order irregular pair. Some irregular primes, such as 157, have more than one pair. See A091887 for a count of the pairs for each irregular prime.

Bernd Kellner, On irregular pairs of higher order (in German)

Cf. A000928, A035112, A091887, A092681.

A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

Original entry on oeis.org

11, 23, 32, 13, 15, 44, 7, 27, 29, 19, 63, 24, 22, 43, 129, 130, 62, 75, 133, 84, 211, 127, 164, 100, 84, 9, 20, 156, 88, 87, 280, 19, 71, 125, 163, 100, 200, 382, 126, 159, 240, 215, 196, 130, 94, 292, 141, 400, 86, 270, 222, 175, 389, 52, 45, 22, 592, 522, 20

Offset: 1

Eric Chen, Dec 26 2014

nonn

A prime p can divide A241601(k) for more than one k; the first few examples are as follows:
p      k
   27, 58
  63, 68
  130, 147
  62, 110
  211, 239
  100, 213
  88, 91, 137
  87, 193, 292
  19, 257
  71, 186, 300
etc.

			19 is the first weak irregular prime and divides A241601(11) = 50521, so a(1) = 11.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A035112, A120337, A128197, A250216.

More terms from Jinyuan Wang, Feb 18 2022

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

A189683 Irregular pairs (p,2k) ordered by increasing k.

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A092681 Least number 2k such that p^2 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

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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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A189684 Number of irregular pairs (p,2n).

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Mathematica

A092682 Least number 2k such that p^3 divides the numerator of the Bernoulli number B(2k), where p is the n-th irregular prime, A000928(n).

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A250215 Least k such that the n-th weak irregular prime (A250216(n)) divides A241601(k).

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Extensions

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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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.