A092230 -id:A092230 - cp's OEIS frontend

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

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

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).

A090789 Even numbers n such that 37^2 (the square of the first irregular prime) divides the numerator of Bernoulli(n).

Original entry on oeis.org

284, 1184, 1616, 2516, 2738, 2948, 3848, 4280, 5180, 5476, 5612, 6512, 6944, 7844, 8214, 8276, 9176, 9608, 10508, 10940, 10952, 11840, 12272, 13172, 13604, 13690, 14504, 14936, 15836, 16268, 16428, 17168, 17600, 18500, 18932, 19166

Offset: 1

T. D. Noe, Feb 26 2004

nonn

Let N(n) be the numerator of the Bernoulli number B(n). This sequence is the union of three arithmetic progressions. The first, n=284+36*37*a, follows from work by Kellner on higher-order irregular pairs. In this case, the second-order pair is (37,284) because n=284 is the smallest even n such that 37^2 | N(n). The second progression, n=37(32+36*b), follows from the first-order pair (37,32). By the Kummer congruence, 37 | N(n) for n=32+36b. By a theorem of Adams, every 37th of these numbers has another factor of 37. The third progression, n=2*37^2c, yields factors of 37^2 by Adams' theorem.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bernd Kellner, On irregular pairs of higher order (in German)
S. S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers
Eric Weisstein's World of Mathematics, Bernoulli Number

Twice A092230.

N:= 20000: # to get all terms <= N
sort(convert({seq(284+36*37*k, k=0..floor((N-284)/36/37)),
seq(1184+36*37*k, k=0..floor((N-1184)/36/37)),
seq(2*37^2*k, k=1..floor(N/2/37^2))},list)); # Robert Israel, Aug 20 2015

nn=10; Union[284+36*37*Range[0, 2nn], 37(32+36*Range[0, 2nn]), 2*37^2*Range[nn]]

These numbers are the union of three arithmetic progressions: 284 + 36*37*k, 32*37 + 36*37*k and 2*37^2*k.

Definition corrected by Robert Israel, Aug 20 2015

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A090789 Even numbers n such that 37^2 (the square of the first irregular prime) divides the numerator of Bernoulli(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions