A247037 -id:A247037 - cp's OEIS frontend

A173955 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4))/16 where Zeta(n, a) is the Hurwitz Zeta function.

0, 1, 58, 7459, 192404, 70791869, 37930481726, 3100675399831, 3000384410275816, 3016572632600497, 512004171837010018, 950047080453398607307, 2104850677799349861903388, 609822785846772474028096357, 611130542819711220012487366

Offset: 1

Artur Jasinski, Mar 03 2010

The denominators are given in A173954.

a(n+2)/A173954(n+2) = (Zeta(2, 3/4) - Zeta(2, n + 7/4))/16 gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(4*n + 3). In the limit n -> infinity the series value is Zeta(2,3/4)/16, with the Hurwitz Zeta function, and it is given in A247037. - Wolfdieter Lang, Nov 15 2017

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A247037.

[0] cat [Numerator((&+[1/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> (Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4))/16:
seq(numer(simplify(r(n))), n=1..15); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[(Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4])/16]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[8*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[1/(4*k + 3)^2, {k, 0, n-2}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)

for(n=1,20, print1(numerator(sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = numerator of r(n) with r(n) = (Pi^2 - 8*Catalan - Zeta(2, n - 1/4))/16, with the Hurwitz Zeta function Z(2, z), and the Catalan constant is given in A006752. With Zeta(2, 3/4) = Pi^2 - 8*Catalan this is the formula given in the name.

Numerator of Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. - G. C. Greubel, Aug 23 2018

Numbers changed according to the old (or new) Mathematica program, and edited by Wolfdieter Lang, Nov 14 2017

Name simplified by Peter Luschny, Nov 14 2017

A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 4, 0, 2, 6, 0, 0, 6, 1, 0, 10, 2, 1, 8, 0, 0, 10, 4, 0, 12, 1, 0, 14, 0, 6, 12, 0, 2, 14, 0, 0, 20, 1, 4, 18, 2, 1, 16, 7, 0, 18, 0, 0, 20, 6, 8, 22, 0, 1, 24, 0, 2, 31, 0, 0, 28, 1, 0, 26, 12, 1, 24, 0, 0, 31, 4, 18, 28, 1, 0, 30, 0, 1

Offset: 1

N. J. A. Sloane, Dec 23 1999

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002131, A050460, A050469, A247037, A326400, A363900.

Cf. A050465, A050466, A050467.

a[n_] := DivisorSum[n, # &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

a(n)={sumdiv(n, d, d*(n/d%4==3))} \\ Andrew Howroyd, Sep 13 2019

G.f.: Sum_{k>=1} k*x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 13 2019
G.f.: Sum_{k>0} x^(4*k-1) / (1 - x^(4*k-1))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050460(n).
a(n) = A050460(n) - A050469(n).
a(n) = (A002131(n) - A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A247037. (End)

Offset changed to 1 by Ilya Gutkovskiy, Sep 13 2019

A282824 Decimal expansion of Pi^2 - 8*K, where K is Catalan's constant.

Original entry on oeis.org

2, 5, 4, 1, 8, 7, 9, 6, 4, 7, 6, 7, 1, 6, 0, 6, 4, 9, 8, 3, 9, 7, 6, 6, 2, 8, 8, 0, 4, 1, 7, 0, 7, 8, 2, 4, 9, 1, 2, 0, 5, 0, 4, 4, 1, 2, 9, 8, 7, 4, 1, 3, 5, 5, 2, 2, 8, 1, 3, 6, 4, 4, 1, 9, 2, 4, 5, 9, 4, 0, 6, 6, 4, 2, 0, 9, 1, 6, 7, 0, 8, 7, 1, 6, 6, 9, 2, 1, 3, 0, 0, 3

Offset: 1

Bruno Berselli, Mar 06 2017

			2.5418796476716064983976628804170782491205044129874135522813644192459406...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95
Eric Weisstein's World of Mathematics, Polygamma Function (formula 25).
Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556

Cf. A000796, A006752, A247037, A173953/A173954, A282823.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)^2 - 8*Catalan(R); // G. C. Greubel, Aug 24 2018

RealDigits[Pi^2 - 8 Catalan, 10, 100][[1]]

Pi^2 - 8*Catalan \\ Charles R Greathouse IV, Jan 31 2018

zetahurwitz(2,3/4) \\ Charles R Greathouse IV, Jan 31 2018

Equals 16*A247037.
Equals Psi(1, 3/4), where Psi(r, x) is the Polygamma function of order r.
Because this equals Zeta(2, 3/4), with the Hurwitz Zeta function, this is the value of the series Sum_{k>=0} 1/(k + 3/4)^2 = 16*Sum_{k>=0} 1/(4*k+3)^2 with partial sums {A173953/(n+2) / A173954(n+2)}{n>=0}. - _Wolfdieter Lang, Nov 14 2017
A282823 - this = 16*A006752. - R. J. Mathar, Jun 07 2024

cp's OEIS Frontend

A173955 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4))/16 where Zeta(n, a) is the Hurwitz Zeta function.

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A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

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A282824 Decimal expansion of Pi^2 - 8*K, where K is Catalan's constant.

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Magma

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