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
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
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Jul 31 2006
STATUS
approved