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A246053 The denominator of h(n-k)*h(k)/h(n) where h(x) = zeta(2*x)*(4^x-2) and k = floor(n/2).

%I A246053 #20 Aug 19 2014 01:09:12
%S A246053 2,2,7,62,381,365,1414477,573370,118518239,5749691557,274638832071,
%T A246053 768018339627,29741482024668555,860983513348179,65053034220152267,
%U A246053 1850237821952082716222,16555640865486520478399,962703047799452264039,29167285342563717499865628061
%N A246053 The denominator of h(n-k)*h(k)/h(n) where h(x) = zeta(2*x)*(4^x-2) and k = floor(n/2).
%C A246053 There are terms that are not squarefree. For example, a(113) is divisible by 631^2 and a(114) is divisible by 103^2. Most terms appear to be divisible by numerator(bernoulli(2*n)/factorial(2*n)) but not all. The first two exceptions are a(1437) and a(23766). - _Hans Havermann_, Aug 16 2014
%H A246053 Hans Havermann, <a href="/A246053/b246053.txt">Table of n, a(n) for n = 0..200</a>
%H A246053 Hans Havermann, <a href="http://chesswanks.com/num/a246053.txt">Factorization table of n, a(n) for n = 0..150</a>
%H A246053 Dinesh S. Thakur, <a href="http://dx.doi.org/10.1090/S0002-9939-2012-11234-1">A note on numerators of Bernoulli numbers</a>, Proc. Amer. Math. Soc. 140 (2012), 3673-3676.
%F A246053 a(n) = A246052(n, floor(n/2)).
%e A246053 a( 0) = 2
%e A246053 a( 1) = 2
%e A246053 a( 2) = 7
%e A246053 a( 3) = 2 * 31
%e A246053 a( 4) = 3 * 127
%e A246053 a( 5) = 5 * 73
%e A246053 a( 6) = 23 * 89 * 691
%e A246053 a( 7) = 2 * 5 * 7 * 8191
%e A246053 a( 8) = 7 * 31 * 151 * 3617
%e A246053 a( 9) = 43867 * 131071
%e A246053 a(10) = 3 * 283 * 617 * 524287
%e A246053 a(11) = 3 * 7 * 11 * 127 * 131 * 337 * 593
%e A246053 a(12) = 3 * 5 * 47 * 103 * 178481 * 2294797
%e A246053 a(13) = 3 * 13 * 31 * 601 * 1801 * 657931
%o A246053 (Sage)
%o A246053 h = lambda x: zeta(2*x)*(4^x-2)
%o A246053 A246053 = lambda n: Integer((h((n+1)//2)*h(n//2)/h(n)).denominator())
%o A246053 [A246053(n) for n in range(19)]
%Y A246053 Cf. A242035 (numerator), A240978 (largest prime divisor), A242050, A246051, A246052.
%K A246053 nonn,frac
%O A246053 0,1
%A A246053 _Peter Luschny_, Aug 12 2014