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%I #31 Mar 12 2015 23:24:04
%S 1,13,19,13,341,9139,43,221,19,270413,1541,667147,79,16211,6479,21437,
%T 103,996151,1,11086933,103759,20033,6533,11341499,51491,8545667,3097,
%U 16211,59,34408161359,1,4137341,5826521,1339,219666403,72719023,223,2977,1501,45423164501,83
%N A027642(6*n+6)/(sequence of period 2:repeat 42,210).
%C Is a(n) always an integer? Is there an a(n) ending with 5?
%C It appears (tested for n <= 800) that a(n) mod 9 is always one of {1, 2, 4, 5, 7, 8}.
%C There is a similar sequence of ratios A027642(10n+1)/(66*A010686(n)) which starts 1, 1, 217, 41, 1, 172081, 71, 697, 4123, 101, 23, 7055321, 131, 2059, 32767, 697, 1, 21896102683,...
%C a(n) is always an integer: 42 = 2*3*7 and 1, 2, and 6 divide 12n+6; 210 = 2*3*5*7 and 1, 2, 4, and 6 divide 12n+12. a(n) never ends in 5 (or 0) since 12n+6 is not divisible by 4 hence the (12n+6)-th Bernoulli denominator is not divisible by 5, and Bernoulli denominators are squarefree and hence the (12n+12)-th Bernoulli denominator, divided by 210, cannot be divisible by 5. - _Charles R Greathouse IV_, Sep 12 2012
%C The previous comments argue that 3 or 5 are never prime divisors of a(n). In addition (tested up to n <=900), 7 apparently is also a non-divisor of a(n). In summary, the prime divisors appear all to be in A140461. - _Jean-François Alcover_, Sep 17 2012
%H Charles R Greathouse IV, <a href="/A216639/b216639.txt">Table of n, a(n) for n = 0..10000</a>
%H A. Joyal, <a href="http://campmath.uqam.ca/deJoyal/bernoulli.pdf">Les nombres de Bernoulli</a> (in French), 2003.
%F a(n) = A027642(6*n+6)/(42*A010686(n)).
%p A010686 := proc(n)
%p op((n mod 2)+1,[1,5]) ;
%p end proc:
%p A216639 := proc(n)
%p A027642(6*n+6)/42/A010686(n) ;
%p end proc: # _R. J. Mathar_, Sep 21 2012
%o (PARI) a(n)=denominator(bernfrac(6*n+6))/if(n%2,210,42) \\ _Charles R Greathouse IV_, Sep 12 2012
%o (PARI) a(n)=my(t=1);fordiv(3*n+3,d,if(isprime(2*d+1),t*=2*d+1));t/if(n%2,105,21) \\ _Charles R Greathouse IV_, Sep 12 2012
%Y Cf. A002445, A027762, A165734, A165949.
%K nonn
%O 0,2
%A _Paul Curtz_, Sep 12 2012
%E a(20)-a(40) from _Charles R Greathouse IV_, Sep 12 2012
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