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 1-4 of 4 results.

A246749 Decimal expansion of F'(rho), an auxiliary constant associated with the asymptotic number of values of the Euler totient function less than a given number, where the function F and the constant rho are defined in A246746.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 02 2014

Keywords

Examples

			5.6977589342301926757529137046852478978581019821783573593459567...

Links

Crossrefs

Cf. A246746.

Programs

  • Mathematica
    digits = 101; F[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; F'[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*k*x^(k - 1), {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 5/10, 6/10}, WorkingPrecision -> digits + 10]; RealDigits[F'[rho], 10, digits] // First

Formula

Let F(x) = sum_{k >= 1} ((k+1)*log(k+1) - k*log(k) - 1)*x^k.
F'(rho), where rho is the unique solution on [0,1) of F(rho)=1,

A234614 Decimal expansion of constant related to the growth of the number of totients.

Original entry on oeis.org

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

Views

Author

Charles R Greathouse IV, Dec 28 2013

Keywords

Comments

Let f_k(x) = x * exp(k (log log log x)^2)/log x. Maier & Pomerance show that, for any e > 0, f_{c-e}(x) << g(x) << f_{c+e}(x) where g(x) gives the number of totients less than x and c is this constant. Loosely, this means f_c(A007617(n)) is about n.

Examples

			0.81781464008363223152559680090296560386485298237899...

Links

Crossrefs

Cf. A007617, A246746.

Programs

  • Mathematica
    digits = 101; F[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 5/10, 6/10}, WorkingPrecision -> digits + 10]; RealDigits[rho, 10, digits] // First ;RealDigits[-1/2/Log[rho],10,90][[1]] (* after Jean-François Alcover at A246746 *)

Formula

See Maier & Pomerance p. 264.
Equals -1/(2*log(c0)), where c0 is a constant whose decimal expansion is A246746. - Amiram Eldar, Jun 19 2018

Extensions

a(8) corrected and more terms added by Amiram Eldar, Jun 19 2018

A246751 Decimal expansion of D, an auxiliary constant associated with the asymptotic number of values of the Euler totient function less than a given number.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 02 2014

Keywords

Examples

			2.176968743559410321739727298735814329767273758965844960238628...

Links

Crossrefs

Cf. A246746, A246749, A234614.

Programs

  • Mathematica
    digits = 99; F[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; F'[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*k*x^(k - 1), {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 5/10, 6/10}, WorkingPrecision -> digits + 10]; c = -1/(2*Log[rho]); d = 2*c*(1 + Log[F'[rho]] - Log[2*c]) - 3/2; RealDigits[d, 10, digits] // First

Formula

Let F(x) = sum_{k >= 1} ((k+1)*log(k+1) - k*log(k) - 1)*x^k.
C = 1/(2*|log(rho)|), where rho is the unique solution on [0,1) of F(rho)=1.
D = 2*C*(1 + log(F'(rho)) - log(2*C)) - 3/2.

A272983 Decimal expansion of the normalized asymptotic mean of omega(m) when m is one of the values <= n taken by Euler's phi totient function.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 12 2016

Keywords

Examples

			2.186263464885754808050867579590101743875879953801252477564664456821...

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.7. Euler Totient Constants, pp. 115-119.

Links

Crossrefs

Cf. A000010, A001221, A246746, A234614, A246751.

Programs

  • Mathematica
    digits = 98; F[x_?NumericQ] := NSum[((k+1)*Log[k+1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 1/2, 3/5}, WorkingPrecision -> digits + 10]; RealDigits[1/(1 - rho), 10, digits] // First

Formula

1/(1 - rho), where rho is A246746.
Showing 1-4 of 4 results.

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