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 1-6 of 6 results.

A065477 Decimal expansion of constant 5 * Pi^2 * A065476 / 48.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Nov 19 2001

Keywords

Examples

			0.743971193350374744686559607585650...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 106.

Links

Crossrefs

Cf. A065476, A078088.

Programs

  • Mathematica
    $MaxExtraPrecision = 500; digits = 98; terms = 500; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{0, 1, 2}, {-2, -6, -2}, terms + 10]]; r[n_Integer] := LR[[n]];  (5 Pi^2/48)*Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)
  • PARI
    (5 * Pi^2 / 48) * prodeulerrat(1 - (p+2)/p^3, 1, 3) \\ Amiram Eldar, Mar 16 2021

A078088 Continued fraction expansion of 5 * Pi^2 * A065476 / 48.

Original entry on oeis.org

0, 1, 2, 1, 9, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 5, 1, 11, 1, 2, 1, 10, 1, 68, 2, 1, 15, 11, 1, 5, 3, 2, 1, 2, 82, 2, 7, 2, 1, 3, 6, 1, 1, 2, 5, 43, 9, 3, 6, 2, 1, 1, 2, 8, 1, 1, 3, 1, 1, 4, 1, 1, 1, 5, 6, 2, 1, 9, 13, 1, 2, 14, 2, 7, 1, 1, 1, 3, 1, 5, 169, 1, 2, 1, 2, 7, 43, 2, 3, 1, 1, 7, 1, 96
Offset: 0

Views

Author

Benoit Cloitre, Dec 02 2002

Keywords

Crossrefs

Cf. A065476, A065477 (decimal expansion).

Programs

  • PARI
    contfrac((5 * Pi^2 / 48) * prodeulerrat(1 - (p+2)/p^3, 1, 3)) \\ Amiram Eldar, Mar 16 2021

Extensions

Corrected from 2nd term on by R. J. Mathar, Sep 19 2008
Offset changed by Andrew Howroyd, Jul 05 2024

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 19 2019

Keywords

Comments

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

Examples

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Crossrefs

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Programs

  • Mathematica
    Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]
  • PARI
    prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Formula

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A330595 Decimal expansion of Product_{primes p} (1 + 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 19 2019

Keywords

Examples

			1.748932997843245303033906997685114802259883493595480897273662144084849...

Crossrefs

Cf. A065464, A065473, A065476, A160889, A328017, A330594, A330596, A338325.

Programs

  • Mathematica
    Do[Print[N[Exp[-Sum[q = Expand[(-p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]
  • PARI
    prodeulerrat(1 + 1/p^2 + 1/p^3) \\ Vaclav Kotesovec, Sep 19 2020

Formula

Equals Sum_{n>=1} 1/A338325(n). - Amiram Eldar, Oct 26 2020

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Dec 19 2019

Keywords

Examples

			1.106960111953217676651179130007439592949548833658122419043134044978777...

Crossrefs

Cf. A065464, A065473, A065476, A208133, A328017, A330595, A330596.

Programs

  • Mathematica
    Do[Print[N[Exp[-Sum[q = Expand[(-p^2 + 2*p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]
  • PARI
    prodeulerrat(1 + 1/p^2 - 2/p^3) \\ Amiram Eldar, Mar 16 2021

A078090 Continued fraction expansion of Product_{p prime >= 3} (1 - (p+2)/p^3).

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 8, 2, 1, 1, 3, 178, 2, 3, 1, 4, 3, 5, 1, 3, 1, 1, 6, 73, 1, 3, 1, 1, 10, 4, 55, 1, 2, 1, 1, 9, 1, 8, 1, 116, 2, 1, 1, 1, 1, 2, 1, 8, 2, 38, 1, 11, 2, 10, 1, 1, 1, 1, 1, 2, 12, 1, 1, 11, 1, 3, 1, 4, 1, 3, 1, 1, 4, 3, 8, 5, 3, 3, 1, 6
Offset: 0

Views

Author

Benoit Cloitre, Dec 02 2002

Keywords

Crossrefs

Cf. A065476 (decimal expansion).

Programs

  • PARI
    contfrac(prodeulerrat(1 - (p+2)/p^3, 1, 3)) \\ Amiram Eldar, Mar 16 2021

Extensions

Offset changed by Andrew Howroyd, Jul 05 2024
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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