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

Author

Alexander Adamchuk, Jul 31 2006

Keywords

• nonn

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.

Examples

			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.
		

Links

• The Bernoulli Number Page, www.bernoulli.org

Crossrefs

Cf. A027641.

Programs

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 *)

Formula

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