A189683 -id:A189683 - cp's OEIS frontend

A189685 Numbers 2k in A189683.

12, 16, 18, 20, 20, 22, 22, 24, 24, 26, 28, 28, 30, 30, 32, 32, 32, 34, 36, 38, 40, 40, 42, 44, 44, 44, 44, 46, 46, 48, 48, 48, 50, 50, 52, 52, 52, 52, 54, 54, 54, 56, 56, 56, 58, 58, 58, 58, 60, 60, 60, 62, 62, 62, 62, 64, 64, 66, 66, 68, 68, 68, 68, 70, 70, 70

Offset: 1

Jonathan Sondow, Apr 25 2011

nonn

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

A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

Original entry on oeis.org

32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, 592, 522, 20, 428, 80, 236, 48, 224, 408, 628, 32, 12, 378, 290, 514, 260, 732, 220, 330, 544, 744, 102

Offset: 1

N. J. A. Sloane

nonn

The ordered pair (p(n),a(n)) where p(n) is the n-th irregular prime is called an irregular pair. Some irregular primes, such as 157, are in more than one pair. See A091887 for the number of pairs for each irregular prime. See A092681 and A092682 for higher-order irregular pairs. - T. D. Noe, Mar 03 2004

			The first irregular prime (37) divides the numerator (-7709321041217) of the 32nd Bernoulli number.

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.

Eric Weisstein's World of Mathematics, Irregular Pair

Cf. A000928, A000367, A035112, A046753, A189683, A189684, A189685.

Do[ p = Prime[ n ]; k = 1; While[ 2*k < p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 3, Print[ 2*k ] ], { n, 3, 200} ]

More terms from Robert G. Wilson v, May 12 2001

A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n).

Original entry on oeis.org

32, 284, 37580, 1072544, 55777784, 325656968, 42764158652, 2444284077476, 46872402575720, 4093248733492712, 167845040875289732, 4841789050865438960, 235423026877046134208, 7818983737604766777920, 95503904455394036720840, 6908622244227620311285724, 114945213060615779807957456

Offset: 1

Bernd C. Kellner and Robert G. Wilson v, Dec 08 2014

nonn

37 is the first irregular prime. The corresponding entry for the second irregular prime 59 is A299466, and for the third irregular prime 67 is A299467.

The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(37,32) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 18 / 19 and 80 / 81. This is caused only by those p-adic digits that are zero.

			a(3) = 37580 because the numerator of B_37580 is divisible by 37^3 and there is no even integer less than 37580 for which this is the case.

Bernd C. Kellner and Robert G. Wilson v, Table of n, a(n) for n = 1..100
Bernd C. Kellner, The Bernoulli Number Page
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.

Cf. 2*A091216, 2*A092230, A189683, A299466, A299467.

p = 37; l = 32; LD = {7, 28, 21, 30, 4, 17, 26, 13, 32, 35, 27, 36, 32, 10, 21, 9, 11, 0, 1, 13, 6, 8, 10, 11, 10, 11, 32, 13, 30, 10, 6, 8, 2, 12, 1, 8, 2, 5, 3, 10, 19, 8, 4, 7, 19, 27, 33, 29, 29, 11, 2, 23, 8, 34, 5, 8, 35, 35, 13, 31, 29, 6, 7, 22, 13, 29, 7, 15, 22, 20, 19, 29, 2, 14, 2, 2, 31, 11, 4, 0, 27, 8, 10, 23, 17, 35, 15, 32, 22, 14, 7, 18, 8, 3, 27, 35, 33, 31, 6}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n - 2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm

Numerator(B_{a(n)}) == 0 (mod 37^n).

Edited for consistency with A299466 and A299467 by Bernd C. Kellner and Jonathan Sondow, Feb 20 2018

A299466 Least even integer k such that numerator(B_k) == 0 (mod 59^n).

Original entry on oeis.org

44, 914, 86464, 8162384, 436993736, 13087518620, 469209221382, 42059215391408, 4083629226737464, 498021221327673308, 5020105038665551466, 1516903461301962815624, 24254443348634296180510, 2604090699795956735657960, 252229046873638875979496022

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 10 2018

nonn

59 is the second irregular prime. The corresponding entry for the first irregular prime 37 is A251782, and for the third irregular prime 67 is A299467.

The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(59,44) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 30 / 31 and 94 / 95. This is caused only by those p-adic digits that are zero.

			a(3) = 86464 because the numerator of B_86464 is divisible by 59^3 and there is no even integer less than 86464 for which this is the case.

Bernd C. Kellner, Table of n, a(n) for n = 1..100
Bernd C. Kellner, The Bernoulli Number Page
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp., 76 (2007), 405-441.
Wikipedia, Irregular pairs

Cf. 2*A091216, 2*A092230, A189683, A251782, A299467.

p = 59; l = 44; LD = {15, 25, 40, 36, 18, 11, 17, 28, 58, 9, 51, 13, 25, 41, 44,17, 43, 35, 21, 10, 21, 38, 9, 12, 40, 43, 45, 30, 41, 0, 3, 25, 34, 49, 45,9, 19, 48, 57, 11, 13, 29, 28, 44, 41, 37, 33, 29, 43, 8, 57, 12, 48, 15,15, 53, 57, 16, 51, 16, 54, 30, 9, 26, 8, 49, 22, 58, 11, 42, 28, 36, 33,45, 24, 32, 18, 12, 29, 45, 40, 27, 19, 40, 41, 11, 42, 49, 35, 41, 57, 54,33, 0, 34, 34, 49, 6, 31}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n -2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm

Numerator(B_{a(n)}) == 0 (mod 59^n).

A299467 Least even integer k such that numerator(B_k) == 0 (mod 67^n).

Original entry on oeis.org

58, 3292, 153640, 12597148, 846312184, 52715297638, 320040068824, 370475739904372, 23170872799129498, 532379740455157312, 111861518490094080436, 1314934469494256636776, 291496130251698265225984, 7852328398132458266800348, 1925603427201316655808983674

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 15 2018

nonn

67 is the third irregular prime. The corresponding entry for the first irregular prime 37 is A251782, and for the second irregular prime 59 is A299466.

The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(67,58) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 22 / 23 and 84 / 85. This is caused only by those p-adic digits that are zero.

			a(3) = 153640 because the numerator of B_153640 is divisible by 67^3 and there is no even integer less than 153640 for which this is the case.

Bernd C. Kellner, Table of n, a(n) for n = 1..100
Bernd C. Kellner, The Bernoulli Number Page
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp., 76 (2007), 405-441.
Wikipedia, Irregular pairs

Cf. 2*A091216, 2*A092230, A189683, A251782, A299466.

p = 67; l = 58; LD = {49, 34, 42, 42, 39, 3, 62, 57, 19, 62, 10, 36, 14, 53, 57, 16, 60, 22, 41, 21, 25, 0, 56, 21, 24, 52, 33, 28, 51, 34, 60, 8, 47, 39, 42, 33, 14, 66, 50, 48, 45, 28, 61, 50, 27, 8, 30, 59, 32, 15, 3, 1, 54, 12, 30, 20, 14, 12, 10, 49, 33, 49, 54, 13, 26, 42, 8, 58, 12, 63, 19, 16, 48, 15, 2, 13, 1, 23, 2, 44, 64, 25, 40, 0, 16, 58, 44, 31, 62, 47, 61, 46, 9, 2, 50, 1, 62, 34, 31}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n - 2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm

Numerator(B_{a(n)}) == 0 (mod 67^n).

A046753 Prime factors of |numerator(B(2n))| which are >= 2n+3.

Original entry on oeis.org

691, 3617, 43867, 283, 617, 131, 593, 103, 2294797, 657931, 9349, 362903, 1721, 1001259881, 37, 683, 305065927, 151628697551, 26315271553053477373, 154210205991661, 137616929, 1897170067619, 1520097643918070802691, 59

Offset: 1

Bill Gosper

nonn

See A189683 for pairs (p,2n) for the primes p in this sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..154 (first 66 terms from T. D. Noe)
Hisanori Mishima, WIFC (World Integer Factorization Center)

Cf. A000367, A000928, A189683, A189684, A189685.

for n do for p in map('first,factor_number(abs(num(bern(2*n))))) do if p>=2*n+3 then (?prin1(p),?prin1('?\-));

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

Definition modified by Jonathan Sondow, Apr 27 2011

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 *)

A190572 A189685/2.

Original entry on oeis.org

6, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 19, 20, 20, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 34, 34, 35, 35, 35

Offset: 1

Jonathan Sondow, May 12 2011

nonn

a(n) = A189683(2n)/2

cp's OEIS Frontend

A189685 Numbers 2k in A189683.

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A035112 Smallest even index 2a such that n-th irregular prime p (A000928(n)) divides Bernoulli_{2a} with 0<=2a<=p-3.

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A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n).

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A299466 Least even integer k such that numerator(B_k) == 0 (mod 59^n).

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A299467 Least even integer k such that numerator(B_k) == 0 (mod 67^n).

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A046753 Prime factors of |numerator(B(2n))| which are >= 2n+3.

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Macsyma

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

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A190572 A189685/2.

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