cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 25 results. Next

A212316 Continued fraction expansion A074760.

Original entry on oeis.org

0, 43, 3, 2, 1, 4, 1, 1, 9, 3, 4, 1, 1, 2, 1, 14, 5, 1, 2, 7, 1, 10, 1, 4, 1, 1, 1, 2, 4, 8, 1, 274, 1, 1, 1, 4, 1, 9, 2, 1, 1, 2, 1, 1, 1, 5, 7, 2, 9, 1, 2, 9, 1, 2, 3, 153, 1, 1, 7, 2, 3, 8, 1, 12, 1, 2, 2, 113, 63, 2, 1, 1, 7, 12, 1, 5, 2
Offset: 0

Views

Author

Edward Jiang, Oct 24 2013

Keywords

Crossrefs

Cf. A074760.

Programs

  • PARI
    contfrac(Euler/2 - (log(Pi))/2 - log(2) + 1)

A104539 Decimal expansion of lambda(2) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Mar 13 2005

Keywords

Examples

			0.0923457352...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.32, p. 204.

Links

Crossrefs

Cf. A074760 (lambda_1), A104540 (lambda_3), A104541 (lambda_4), A104542 (lambda_5).
Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

Programs

  • Mathematica
    lambda[n_] := Limit[D[s^(n - 1)*Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; Join[{0}, RealDigits[lambda[2], 10, 102] // First]
lambda[2] = 1 + EulerGamma - EulerGamma^2 + Pi^2/8 - Log[4 Pi] - 2*StieltjesGamma[1]; Join[{0}, RealDigits[lambda[2], 10, 102] // First] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + e - e^2 + Pi^2/8 - 2 g[1] - Log[4 Pi]], 10, 110, -1][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

A104540 Decimal expansion of lambda(3) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Mar 13 2005

Keywords

Examples

			0.207638920...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.32, p. 204.

Links

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104541 (lambda_4), A104542 (lambda_5).
Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

Programs

  • Mathematica
    lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[3], 110]][[1]][[1 ;; 102]] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 3 e/2 - 3 e^2 + e^3 + 3 Pi^2/8 - 6 g[1] + 3 e g[1] + 3 g[2]/2 - Log[8] - 3 Log[Pi]/2 - 7 Zeta[3]/8], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

Formula

lambda(3) = 3*Pi^2/8 - 3*log(2) - 3*log(Pi)/2 + 3*gamma/2 - 3*gamma^2 + gamma^3 + 3*gamma*gamma(1) - 6*gamma(1) + 3*gamma(2)/2 - 7*zeta(3)/8 + 1. - Jean-François Alcover, Jul 02 2014

A104541 Decimal expansion of lambda(4) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Mar 13 2005

Keywords

Examples

			0.368790479...

Links

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104542 (lambda_5).
Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

Programs

  • Mathematica
    lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[4], 110]][[1]][[1 ;; 102]] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 2 e - 6 e^2 + 4 e^3 - e^4 + 3 Pi^2/4 + Pi^4/96 - 12 g[1] + 12 e g[1] - 4 e^2 g[1] - 2 g[1]^2 + 6 g[2] - 2 e g[2] - 2 g[3]/3 - 2 Log[4 Pi] - 7 Zeta[3]/2], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

Formula

3*Pi^2/4 + Pi^4/96 - 2*log(4) - 2*log(Pi) + 2*gamma - 6*gamma^2 + 4*gamma^3 - gamma^4 - 12*gamma(1) + 12*gamma*gamma(1) - 4*gamma^2*gamma(1) - 2*gamma(1)^2 + 6*gamma(2) - 2*gamma*gamma(2) - 2*gamma(3)/3 - 7*zeta(3)/2 + 1. - Jean-François Alcover, Jul 02 2014

A104542 Decimal expansion of lambda(5) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Mar 13 2005

Keywords

Examples

			0.575542714...

Links

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).
Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

Programs

  • Mathematica
    lambda[n_] := Limit[D[s^(n - 1)*Log[xi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[5], 110]][[1]][[1 ;; 102]] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 5 e/2 - 10 e^2 + 10 e^3 - 5 e^4 + e^5 + 5 Pi^2/4 + (5 Pi^4)/96 - 20 g[1] + 30 e g[1] - 20 e^2 g[1] + 5 e^3 g[1] - 10 g[1]^2 + 5 e g[1]^2 + 15 g[2] - 10 e g[2] + 5/2 e^2 g[2] + 5/2 g[1] g[2] - 10 g[3]/3 + 5/6 e g[3] + 5 g[4]/24 - Log[32] - 5 Log[Pi]/2 - 35 Zeta[3]/4 - 31 Zeta[5]/32], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

Formula

lambda(5) = 5*Pi^2/4 + 5*Pi^4/96 - 5*log(4)/2 - 5*log(Pi)/2 + 5*gamma/2 - 10*gamma^2 + 10*gamma^3 - 5*gamma^4+gamma^5 - 20*gamma(1) + 30*gamma*gamma(1) - 20*gamma^2*gamma(1) + 5*gamma^3*gamma(1) - 10*gamma(1)^2 + 5*gamma*gamma(1)^2 + 15*gamma(2) - 10*gamma*gamma(2) + 5/2*gamma^2*gamma(2) + 5/2*gamma(1)*gamma(2) - 10*gamma(3)/3 + 5/6*gamma*gamma(3) + 5*gamma(4)/24 - 35*zeta(3)/4 - 31*zeta(5)/32+1. - Jean-François Alcover, Jul 02 2014

A306339 Decimal expansion of lambda(6) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Feb 08 2019

Keywords

Examples

			0.8275660122823792974...

Links

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).
Cf. A104542 (lambda_5), A306340 (lambda_7), A306341 (lambda_8).

Programs

  • Mathematica
    RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 3 e - 15 e^2 + 20 e^3 - 15 e^4 + 6 e^5 - e^6 + 15 Pi^2/8 + 5 Pi^4/32 + Pi^6/960 - 30 g[1] + 60 e g[1] - 60 e^2 g[1] + 30 e^3 g[1] - 6 e^4 g[1] - 30 g[1]^2 + 30 e g[1]^2 - 9 e^2 g[1]^2 - 2 g[1]^3 + 30 g[2] - 30 e g[2] + 15 e^2 g[2] - 3 e^3 g[2] + 15 g[1] g[2] - 6 e g[1] g[2] - 3 g[2]^2/4 - 10 g[3] + 5 e g[3] - e^2 g[3] - g[1] g[3] + 5 g[4]/4 - e g[4]/4 - g[5]/20 - 3 Log[4 Pi] - 35 Zeta[3]/2 - 93 Zeta[5]/16], 10, 110][[1]]

A306340 Decimal expansion of lambda(7) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Feb 08 2019

Keywords

Examples

			1.124460117570959490...

Links

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3).
Cf. A104541 (lambda_4), A104542 (lambda_5), A306339 (lambda_6), A306341 (lambda_8).

Programs

  • Mathematica
    RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 7 e/2 - 21 e^2 + 35 e^3 - 35 e^4 + 21 e^5 - 7 e^6 + e^7 + 21 Pi^2/8 + 35 Pi^4/96 + 7 Pi^6/960 - 42 g[1] + 105 e g[1] - 140 e^2 g[1] + 105 e^3 g[1] - 42 e^4 g[1] + 7 e^5 g[1] - 70 g[1]^2 + 105 e g[1]^2 - 63 e^2 g[1]^2 + 14 e^3 g[1]^2 - 14 g[1]^3 + 7 e g[1]^3 + 105 g[2]/2 - 70 e g[2] + 105/2 e^2 g[2] - 21 e^3 g[2] + 7/2 e^4 g[2] + 105/2 g[1] g[2] - 42 e g[1] g[2] + 21/2 e^2 g[1] g[2] + 7/2 g[1]^2 g[2] - 21 g[2]^2/4 + 7/4 e g[2]^2 - 70 g[3]/3 + 35/2 e g[3] - 7 e^2 g[3] + 7/6 e^3 g[3] - 7 g[1] g[3] + 7/3 e g[1] g[3] + 7/12 g[2] g[3] + 35 g[4]/8 - 7/4 e g[4] + 7/24 e^2 g[4] + 7/24 g[1] g[4] - 7 g[5]/20 + 7/120 e g[5] + 7 g[6]/720 - 7/2 Log[4 Pi] - 245 Zeta[3]/8 - 651 Zeta[5]/32 - 127 Zeta[7]/128], 10, 110][[1]]

A306341 Decimal expansion of lambda(8) in Li's criterion.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Feb 08 2019

Keywords

Examples

			1.465755677147060632...

Links

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).
Cf. A104542 (lambda_5), A306339 (lambda_6), A306340 (lambda_7).

Programs

  • Mathematica
    RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 4 e - 28 e^2 + 56 e^3 - 70 e^4 + 56 e^5 - 28 e^6 + 8 e^7 - e^8 + 7 Pi^2/2 + 35 Pi^4/48 + 7 Pi^6/240 + 17 Pi^8/161280 - 56 g[1] + 168 e g[1] - 280 e^2 g[1] + 280 e^3 g[1] - 168 e^4 g[1] + 56 e^5 g[1] - 8 e^6 g[1] - 140 g[1]^2 + 280 e g[1]^2 - 252 e^2 g[1]^2 + 112 e^3 g[1]^2 - 20 e^4 g[1]^2 - 56 g[1]^3 + 56 e g[1]^3 - 16 e^2 g[1]^3 - 2 g[1]^4 + 84 g[2] - 140 e g[2] + 140 e^2 g[2] - 84 e^3 g[2] + 28 e^4 g[2] - 4 e^5 g[2] + 140 g[1] g[2] - 168 e g[1] g[2] + 84 e^2 g[1] g[2] - 16 e^3 g[1] g[2] + 28 g[1]^2 g[2] - 12 e g[1]^2 g[2] - 21 g[2]^2 + 14 e g[2]^2 - 3 e^2 g[2]^2 - 2 g[1] g[2]^2 - 140 g[3]/3 + 140/3 e g[3] - 28 e^2 g[3] + 28/3 e^3 g[3] - 4/3 e^4 g[3] - 28 g[1] g[3] + 56/3 e g[1] g[3] - 4 e^2 g[1] g[3] - 4/3 g[1]^2 g[3] + 14/3 g[2] g[3] - 4/3 e g[2] g[3] - g[3]^2/9 + 35 g[4]/3 - 7 e g[4] + 7/3 e^2 g[4] - 1/3 e^3 g[4] + 7/3 g[1] g[4] - 2/3 e g[1] g[4] - 1/6 g[2] g[4] - 7 g[5]/5 + 7/15 e g[5] - 1/15 e^2 g[5] - 1/15 g[1] g[5] + 7 g[6]/90 - 1/90 e g[6] - g[7]/630 - 4 Log[4 Pi] - 49 Zeta[3] - 217 Zeta[5]/4 - 127 Zeta[7]/16], 10, 110][[1]]

A245275 Decimal expansion of sum_{r in Z}(1/r^2) where Z is the set of all nontrivial zeros r of the Riemann zeta function.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jul 16 2014

Keywords

Examples

			-0.046154317295804602757107990379077303530267962324144990348848453508...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.21 Stieltjes Constants, p. 168.

Links

Crossrefs

Cf. A082633, A074760, A195423, A245276.

Programs

  • Mathematica
    Join[{0}, RealDigits[-Pi^2/8 + EulerGamma^2 + 2*StieltjesGamma[1] + 1, 10, 104] // First]
  • PARI
    -Pi^2/8+Euler^2+1+2*intnum(x=0,oo,(1/tanh(Pi*x)-1)*(x*log(1+x^2)-2*atan(x))/(2*(1+x^2))) \\ Charles R Greathouse IV, Mar 10 2016

Formula

-Pi^2/8 + gamma^2 + 2*gamma(1) + 1, where gamma is Euler's constant and gamma(1) is the first Stieltjes constant.

A245276 Decimal expansion of sum_{r in Z}(1/r^3) where Z is the set of all nontrivial zeros r of the Riemann zeta function.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jul 16 2014

Keywords

Examples

			-0.000111158231452105922762668238914578473964189248986518770273452672891213...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.21 Stieltjes Constants, p. 168.

Links

Crossrefs

Cf. A074760, A195423, A245275.

Programs

  • Mathematica
    Join[{0, 0, 0}, RealDigits[-7*Zeta[3]/8 + EulerGamma^3 + 3*EulerGamma*StieltjesGamma[1] + 3/2*StieltjesGamma[2] + 1, 10, 103] // First]

Formula

-7*zeta(3)/8 + gamma^3 + 3*gamma*gamma(1) + 3/2*gamma(2) + 1, where gamma is Euler's constant and gamma(n) is the n-th Stieltjes constant.
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