A119864 Numbers n such that the numerator of BernoulliB[n] is divisible by 691.

12, 200, 702, 890, 1382, 1392, 1580, 2082, 2270, 2764, 2772, 2960, 3462, 3650, 4146, 4152, 4340, 4842, 5030, 5528, 5532, 5720, 6222, 6410, 6910, 6912, 7100, 7602, 7790, 8292, 8480, 8982, 9170, 9672, 9674, 9860, 10362, 10550, 11052, 11056, 11240, 11742

Offset

1,1

Comments

a(n) is a union of 3 arithmetic progressions: 12 + 690*n = {12,702,1392,2082,2772,3462,4152,4842,5532,6222,6912,7602,8292,8982,9672,...}, 200 + 690*n = {200,890,1580,2270,2960,3650,4340,5030,5720,6410,7100,7790,8480,9170,9860,...}, 2*691*n = {1382,2764,4146,5528,6910,8292,9674,...}. Note that Numerator[BernoulliB[8292]] is divisible by 691^2, where a(n) = 8292 = 12 + 690*13 = 691*12. It appears that Numerator[BernoulliB[138200]] is also divisible by 691^2 because a(n) = 138200 = 200 + 690*201 = 691*200.

Links

Table of n, a(n) for n=1..42.

The Bernoulli Number Page, www.bernoulli.org

Formula

Mod[ Numerator[ BernoulliB[ a(n) ]], 691] = 0.

Example

BernoulliB[n] sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...
a(1) = 12 because Numerator[BernoulliB[12]] = 691.

Mathematica

Select[Union[Table[2n*691, {n, 1, 30}], Table[12+690*n, {n, 0, 30}], Table[200+690*n, {n, 0, 30}]], #<=20000&]
Select[Range[2, 12000, 2], Divisible[Numerator[BernoulliB[#]], 691]&] (* Harvey P. Dale, Nov 19 2014 *)

Crossrefs

Cf. A027641.

Sequence in context: A115865 A291285 A159359 * A036240 A346509 A355127

Adjacent sequences: A119861 A119862 A119863 * A119865 A119866 A119867

Keyword

nonn

Author

Alexander Adamchuk, Jul 31 2006

Status

approved