A222183 -id:A222183 - cp's OEIS frontend

A173947 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.

0, 16, 416, 34096, 5794624, 1680121936, 82501802464, 2065646660464, 1739147340740224, 210617970218777104, 288533264855755545376, 485294472126860897387056, 485518650207447822251456

Offset: 0

Artur Jasinski, Mar 03 2010

For A173947/16 see A173949.

a(n+1)/A173948(n+1) =: r(n) = (Zeta(2, 1/4) - Zeta(2, n + 5/4)), the partial sum Sum_{k=0..n} 1/(k + 1/4)^2, n >= 0. The limit is Zeta(2, 1/4) = A282823 = 16*A222183. - Wolfdieter Lang, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Trigamma Function

Cf. A006752, A120268, A173945, A173948 (denominators), A173949.

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

r := n -> Psi(1, 1/4) - Zeta(0, 2, n+1/4):
seq(numer(simplify(r(n))), n=0..13); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4]]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[128*n*Sum[(1 + 4*k + 2*n) / ((1 + 4*k)^2*(1 + 4*k + 4*n)^2), {k, 0, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[16*Sum[1/(4*k + 1)^2, {k, 0, n - 1} ], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)

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

a(n) = numerator of 8*Catalan + Pi^2 - Zeta(2, (4 n + 1)/4), with the Catalan constant given in A006752.

a(n) = numerator(r(n)) with r(n) = Zeta(2, 1/4) - Zeta(2, n + 1/4), with the Hurwitz Zeta function (see the name). With Zeta(2, 1/4) = Psi(1, 1/4) = 8*Catalan + Pi^2 this is the preceding formula, where Psi(1, z) is the Trigamma function. - Wolfdieter Lang, Nov 14 2017

Name simplified and offset set to 0 by Peter Luschny, Nov 14 2017

A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function.

Original entry on oeis.org

0, 1, 26, 2131, 362164, 105007621, 5156362654, 129102916279, 108696708796264, 13163623138673569, 18033329053484721586, 30330904507928806086691, 30344915637965488890716, 1487479897654682071525709

Offset: 0

Artur Jasinski, Mar 03 2010

For the Catalan constant see A006752.
The denominators are given in A173948.
a(n+1)/A173948(n+1), for n>= 0, gives the partial sum Sum_{k=0..n} 1/(4*k + 1)^2. For {(4*k + 1)^2}A016814.%20The%20limit%20n%20-%3E%20infinity%20is%20given%20in%20A222183%20as%201.074833072...%20.%20-%20_Wolfdieter%20Lang">{k>=0} see A016814. The limit n -> infinity is given in A222183 as 1.074833072... . - _Wolfdieter Lang, Nov 14 2017

			The rationals r(n) begin: 0/1, 1/1, 26/25, 2131/2025, 362164/342225, 105007621/98903025, 5156362654/4846248225, 129102916279/121156205625, 108696708796264/101892368930625, 13163623138673569/12328976640605625, ... - _Wolfdieter Lang_, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Polygamma Function

Cf. A016814, A120268, A173945, A173947, A173948, A222183.

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

r := n -> (Psi(1, 1/4) - Zeta(0, 2, n+1/4))/16:
seq(numer(simplify(r(n))), n=0..13); # Peter Luschny, Nov 14 2017

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

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

a(n) = numerator of expression (8*Catalan + Pi^2 - Zeta(2, (4*n + 1)/4))/16.

a(n) = numerator(r(n)) with r(n) = (Zeta(2,1/4) - Zeta(2, n + 1/4))/16, with the Hurwitz Zeta function Z(2, k). With Zeta(2,1/4) = 8 Catalan + Pi^2 this is the preceding formula, and Zeta(2, n + 1/4) = Psi(1, n + 1/4) with the polygamma (trigamma) function Psi(1, k). - Wolfdieter Lang, Nov 14 2017

Edited by Wolfdieter Lang, Nov 14 2017

Name changed according to a formula of Lang by Peter Luschny, Nov 14 2017

A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 10, 12, 11, 12, 14, 14, 18, 16, 18, 20, 19, 24, 22, 22, 23, 24, 31, 28, 30, 28, 30, 36, 31, 32, 34, 36, 42, 40, 38, 38, 42, 48, 42, 44, 43, 44, 60, 46, 47, 48, 50, 62, 54, 56, 54, 60, 66, 56, 58, 60, 59, 72, 62, 62, 73, 64, 84, 68

Offset: 1

N. J. A. Sloane, Dec 23 1999

Not multiplicative: a(3)*a(7) <> a(21), for example.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001826, A002131, A050449, A050464, A050469, A222183.

Cf. A050461, A050462, A050463.

A050460 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050460(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#&]; Array[a, 70] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=sumdiv(n,d,if(n/d%4==1,d)) \\ Charles R Greathouse IV, Dec 04 2013

G.f.: Sum_{n>0} n*x^n/(1-x^(4*n)). - Vladeta Jovovic, Nov 14 2002
G.f.: Sum_{k>0} x^(4*k-3) / (1 - x^(4*k-3))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050464(n).
a(n) = A050469(n) + A050464(n).
a(n) = (A002131(n) + A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A222183. (End)

A282823 Decimal expansion of Pi^2 + 8*K, where K is Catalan's constant.

Original entry on oeis.org

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

Offset: 2

Bruno Berselli, Mar 06 2017

			17.19732915450711073927131911933522402150689440149416770054533433319414...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Polygamma Function (formula 24).
Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556.

Cf. A000796, A006752, A222183, A282824.

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

Psi(1,1/4) ; evalf(%) # R. J. Mathar, Dec 04 2024

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

Pi^2 + 8*Catalan \\ Charles R Greathouse IV, Mar 04 2018

zetahurwitz(2,1/4) \\ Charles R Greathouse IV, Mar 04 2018

Equals 16*A222183.
Equals Psi(1, 1/4), where Psi(r, x) is the Polygamma function of order r.
Equals Sum_{k>=0} 1/(k + 1/4)^2. - Amiram Eldar, May 17 2022

A247037 Decimal expansion of Sum_{k >= 0} 1/(4*k+3)^2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 10 2014

			0.158867477979475406149853930026067390570031525811713347...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.7 Catalan's constant p. 55.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A006752, A173955/A173954, A222068, A222183.

SetDefaultRealField(RealField(100)); R:= RealField(); (Pi(R)^2 - 8*Catalan(R))/16; // G. C. Greubel, Aug 24 2018

RealDigits[Pi^2/16 - Catalan/2 , 10, 103] // First

Pi^2/16 - Catalan/2 \\ Charles R Greathouse IV, Jan 30 2018

zetahurwitz(2,3/4)/16 \\ Charles R Greathouse IV, Jan 30 2018

sumpos(k=0,1/(4*k+3)^2) \\ Charles R Greathouse IV, Jan 30 2018

Equals Pi^2/16 - G/2, where G is Catalan's constant.
Equals A222068 - A006752/2.
Equals zeta(2, 3/4)/16 = Psi(1, 3/4)/16, with the Hurwitz zeta function and the Trigamma function Psi(1, z), and the partial sums of the series given in the name are {A173955(n+2) / A173954(n+2)}{n>=0}. - _Wolfdieter Lang, Nov 14 2017
Equals Integral_{x=1..oo} log(x)/(x^4 - 1) dx. - Amiram Eldar, Jul 17 2020

cp's OEIS Frontend

A173947 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.

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A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function.

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A050460 a(n) = Sum_{d|n, n/d=1 mod 4} d.

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A282823 Decimal expansion of Pi^2 + 8*K, where K is Catalan's constant.

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A247037 Decimal expansion of Sum_{k >= 0} 1/(4*k+3)^2.

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