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%I #36 Sep 08 2022 08:45:51
%S 1,1,25,2025,342225,98903025,4846248225,121156205625,101892368930625,
%T 12328976640605625,16878369020989100625,28372538324282678150625,
%U 28372538324282678150625,1390254377889851229380625,3905224547492592103330175625,1409786061644825749302193400625,5245813935380396613153461643725625
%N a(n) = denominator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.
%C Presumably conjectures:
%C For n>=2 numbers in this sequence are divisible by 25.
%C For n>=7 numbers in this sequence are divisible by 25^2.
%H G. C. Greubel, <a href="/A173948/b173948.txt">Table of n, a(n) for n = 0..250</a>
%F a(n) = denominator of 8*Catalan + Pi^2 - Zeta(2, (4*n + 1)/4), with the Hurwitz Zeta function, and Catalan is given in A006752. [See the name with Zeta(2, 1/4) = Psi(1, 1/4) = 8*Catalan + Pi^2, and the Trigamma function Psi(1, z).]
%p r := n -> Psi(1, 1/4) - Zeta(0, 2, n+1/4):
%p seq(denom(simplify(r(n))), n=0..16); # _Peter Luschny_, Nov 14 2017
%t Table[Denominator[FunctionExpand[8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4]]], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t Denominator[Table[Sum[1/(4*k + 1)^2, {k, 0, n-1} ], {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%o (PARI) for(n=0,20, print1(denominator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ _G. C. Greubel_, Aug 22 2018
%o (Magma) [1] cat [Denominator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // _G. C. Greubel_, Aug 22 2018
%Y Cf. A006752, A120268, A173945, A173947 (numerators).
%K frac,nonn
%O 0,3
%A _Artur Jasinski_, Mar 03 2010
%E Name simplified by _Peter Luschny_, Nov 14 2017
%E Formula reformulated. - _Wolfdieter Lang_, Nov 14 2017.
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