A173948 a(n) = denominator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.

1, 1, 25, 2025, 342225, 98903025, 4846248225, 121156205625, 101892368930625, 12328976640605625, 16878369020989100625, 28372538324282678150625, 28372538324282678150625, 1390254377889851229380625, 3905224547492592103330175625, 1409786061644825749302193400625, 5245813935380396613153461643725625

Offset: 0

Author

Artur Jasinski, Mar 03 2010

Keywords

• frac
• nonn

Comments

Presumably conjectures:

For n>=2 numbers in this sequence are divisible by 25.

For n>=7 numbers in this sequence are divisible by 25^2.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A006752, A120268, A173945, A173947 (numerators).

Programs

Magma

[1] cat [Denominator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 22 2018

Maple

r := n -> Psi(1, 1/4) - Zeta(0, 2, n+1/4):
seq(denom(simplify(r(n))), n=0..16); # Peter Luschny, Nov 14 2017

Mathematica

Table[Denominator[FunctionExpand[8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4]]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Denominator[Table[Sum[1/(4*k + 1)^2, {k, 0, n-1} ], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)

PARI

for(n=0,20, print1(denominator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018

Formula

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).]

Extensions

Name simplified by Peter Luschny, Nov 14 2017

Formula reformulated. - Wolfdieter Lang, Nov 14 2017.