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.

A256784 Decimal expansion of the generalized Euler constant gamma(5,12) (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			-0.0033729493224032970250324948185921946160340346994983953873167...

Links

Crossrefs

Cf. A001620 (EulerGamma), A016635, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/12 + 1/24*(Pi(R)*(2-Sqrt(3)) + 2*(Sqrt(3)+1)*Log(2) + Log(3) - 4*Sqrt(3)*Log(Sqrt(3)+1)); // G. C. Greubel, Aug 27 2018
  • Mathematica
    Join[{0, 0}, RealDigits[-Log[12]/12 - PolyGamma[5/12]/12, 10, 105] // First]
  • PARI
    default(realprecision, 100); Euler/12 + 1/24*(Pi*(2-sqrt(3)) + 2*(sqrt(3)+1)*log(2) + log(3) - 4*sqrt(3)*log(sqrt(3)+1)) \\ G. C. Greubel, Aug 27 2018

Formula

Equals EulerGamma/12 + 1/24*(Pi*(2-sqrt(3)) + 2*(sqrt(3)+1)*log(2) + log(3) - 4*sqrt(3) * log(sqrt(3)+1)).
Equals -(psi(5/12) + log(12))/12. - Amiram Eldar, Jan 07 2024

A234518 Decimal expansion of log_12 (28/13).

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Jan 03 2014

Keywords

Comments

Decimal expansion of maximal value of function alpha(n) = alpha-deviation from primality of number n = log_n(sigma(n)) - log_n(n+1) = log_n[sigma(n) / (n+1)] for n = 12, when alpha(12) = log_12(sigma(12)) - log_12(12+1) = log_12(28) - log_12(13) = log_12 (28/13) = 0,308766187…; alpha(p) = 0 for p = prime.

Examples

			0,3087661875664928997884010546628878661481631771557148…

Links

Crossrefs

Cf. A234515, A234516, A234517, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

  • Mathematica
    RealDigits[Log[12,28/13],10,120][[1]] (* Harvey P. Dale, Apr 17 2022 *)
  • PARI
    log(28/13)/log(12) \\ Michel Marcus, Dec 11 2014

Formula

Decimal expansion of (A016651-A016636) / A016635.

A256783 Decimal expansion of the generalized Euler constant gamma(1,12).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			0.83024988988662433938903419703214965055579639279727496201543...

Links

Crossrefs

Cf. A001620 (EulerGamma), A016635, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/12 + (1/24)*(Pi(R)*(2+Sqrt(3)) - 2*(Sqrt(3)-1)*Log(2) + Log(3) + 4*Sqrt(3)*Log(Sqrt(3)+1)); // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[-Log[12]/12 - PolyGamma[1/12]/12, 10, 103] // First
  • PARI
    default(realprecision, 100); Euler/12 + 1/24*(Pi*(2+sqrt(3)) - 2*(sqrt(3)-1)*log(2) + log(3) + 4*sqrt(3)*log(sqrt(3)+1)) \\ G. C. Greubel, Aug 28 2018

Formula

Equals EulerGamma/12 + 1/24*(Pi*(2+sqrt(3)) - 2*(sqrt(3)-1)*log(2) + log(3) + 4*sqrt(3) * log(sqrt(3)+1)).
Equals Sum_{n>=0} (1/(12n+1) - 1/12*log((12n+13)/(12n+1))).
Equals -(psi(1/12) + log(12))/12. - Amiram Eldar, Jan 07 2024

A016740 Continued fraction for log(12).

Original entry on oeis.org

2, 2, 16, 15, 1, 2, 1, 1, 1, 16, 1, 12, 1, 2, 1, 6, 1, 6, 4, 3, 1, 4, 10, 3, 1, 1, 28, 1, 1, 1, 1, 2, 4, 3, 1, 2, 1, 1, 25, 3, 1, 44, 1, 3, 1, 25, 1, 17, 7, 15, 7, 15, 1, 3, 1, 2, 1, 1, 2, 7, 1, 1, 1, 4, 1, 16, 1, 4, 6, 1, 1, 1, 1, 12, 4, 7, 14, 11, 1, 1, 1, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			2.4849066497880003102297094... = 2 + 1/(2 + 1/(16 + 1/(15 + 1/(1 + ...)))). - _Harry J. Smith_, May 16 2009

Links

Crossrefs

Cf. A016635 (decimal expansion).

Programs

  • Magma
    ContinuedFraction(Log(12)); // G. C. Greubel, Sep 15 2018
  • Mathematica
    ContinuedFraction[Log[12], 100] (* G. C. Greubel, Sep 15 2018 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(12)); for (n=1, 20000, write("b016740.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

Extensions

Offset changed by Andrew Howroyd, Jul 10 2024

A380005 Decimal expansion of (7/3)*log(log(12)) - exp(gamma)*log(log(12))^2, where gamma is the Euler-Mascheroni constant (A001620).

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Jan 14 2025

Keywords

Comments

Theorem 2 in Robin (1984) states that, for n >= 3, sigma(n)/n <= exp(gamma)*log(log(n)) + c/log(log(n)), with equality for n = 12, where sigma is the sum-of-divisors function (A000203) and c is the constant given by the present sequence. Cf. also Weisstein, eqs. (29) - (33).

Examples

			0.64821364942179976272009425643532901899304479911015...

References

  • G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, Journal de Mathématiques Pures et Appliquées, 63 (1984), pp. 187-213 (in French). See A073004 for a scanned copy.

Links

Crossrefs

Cf. A000203, A001620, A016635, A058209, A073004.

Programs

  • Mathematica
    First[RealDigits[7/3*# - Exp[EulerGamma]*#^2, 10, 100]] & [Log[Log[12]]]

Formula

Equals (7/3)*log(A016635) - A073004*log(A016635)^2.

A323458 Decimal expansion of log(2^(1/2)*3^(1/3) / 6^(1/6)).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jan 20 2019

Keywords

Examples

			0.4141511082980000517049515799731464734664151377572...

Links

Crossrefs

Suggested by A230191.
Cf. A002162, A002391, A016635.
Cf. A001008, A002805.

Programs

  • Mathematica
    RealDigits[Log[2^(1/2)*3^(1/3) / 6^(1/6)], 10, 101][[1]] (* Georg Fischer, Apr 04 2020 *)
  • PARI
    log( 2^(1/2)*3^(1/3) / 6^(1/6) ) \\ Charles R Greathouse IV, May 15 2019

Formula

From Jianing Song, Jan 23 2019: (Start)
Equals (1/6)*log(12) = (1/6)*A016635.
Equals (1/3)*log(2) + (1/6)*log(3) = (1/3)*A002162 + (1/6)*A002391. (End)
Equals Sum_{k>=1} H(2*k-1)/4^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, May 30 2021

Extensions

a(99) corrected by Georg Fischer, Apr 04 2020
Showing 1-6 of 6 results.

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

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