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%I #4 Nov 19 2014 16:44:48
%S 12,200,702,890,1382,1392,1580,2082,2270,2764,2772,2960,3462,3650,
%T 4146,4152,4340,4842,5030,5528,5532,5720,6222,6410,6910,6912,7100,
%U 7602,7790,8292,8480,8982,9170,9672,9674,9860,10362,10550,11052,11056,11240,11742
%N Numbers n such that the numerator of BernoulliB[n] is divisible by 691.
%C 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.
%H <a href="http://www.bernoulli.org/">The Bernoulli Number Page</a>, www.bernoulli.org
%F Mod[ Numerator[ BernoulliB[ a(n) ]], 691] = 0.
%e 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, ...
%e a(1) = 12 because Numerator[BernoulliB[12]] = 691.
%t Select[Union[Table[2n*691,{n,1,30}],Table[12+690*n,{n,0,30}],Table[200+690*n,{n,0,30}]], #<=20000&]
%t Select[Range[2,12000,2],Divisible[Numerator[BernoulliB[#]],691]&] (* _Harvey P. Dale_, Nov 19 2014 *)
%Y Cf. A027641.
%K nonn
%O 1,1
%A _Alexander Adamchuk_, Jul 31 2006
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