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 1-7 of 7 results.

A256683 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-6,-5].

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 08 2015

Keywords

Examples

			Gamma(-5.66716244155688553584947417451815542471179578769484889367...)
= 0.00932459448261485052171192379918266310927330588790814482...

Links

Crossrefs

Cf. A030169, A030171, A175472, A175473, A256681, A256682, A256684-A256687.

Programs

  • Mathematica
    digits = 104; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -11/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Formula

Solution to PolyGamma(x) = 0 in the interval [-6,-5]

A256685 Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-8,-7].

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 08 2015

Keywords

Examples

			Gamma(-7.6877883250316260374400988918437049536838217978826433594...)
= 0.0001818784449094041881014174426244626530404358160668...

Links

Crossrefs

Cf. A030169, A030171, A175472, A175473, A256681-A256684, A256686, A256687.

Programs

  • Mathematica
    digits = 106; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -15/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Formula

Solution to PolyGamma(x) = 0 in the interval [-8,-7].

A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 03 2021

Keywords

Comments

The sum is Sum_{k>=0} 1/x_k^2, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

Examples

			5.26798012435239798373562163629331979431626684387002...

Links

Crossrefs

Cf. A344965, A344966, A344967, A344968.
Cf. A001620, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

Programs

  • Mathematica
    RealDigits[Pi^2/2 + EulerGamma^2, 10, 100][[1]]

Formula

Equals Pi^2/2 + gamma^2 = A102753 + A155969, where gamma is Euler's constant (A001620).

A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 03 2021

Keywords

Comments

The sum is Sum_{k>=0} 1/x_k^3, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

Examples

			-7.84898826280450630489883732716055067110164127911638...

Links

Crossrefs

Cf. A344964, A344966, A344967, A344968.
Cf. A001620, A002117, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

Programs

  • Mathematica
    RealDigits[EulerGamma*Pi^2/2 + 4*Zeta[3]  + EulerGamma^3, 10, 100][[1]]

Formula

Equals -gamma*Pi^2/2 - 4*zeta(3) - gamma^3, where gamma is Euler's constant (A001620).

A344966 Decimal expansion of the sum of the reciprocals of the fourth powers of the zeros of the digamma function.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 03 2021

Keywords

Comments

The sum is Sum_{k>=0} 1/x_k^4, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

Examples

			15.90184703322349156972084557358425176519256672643402...

Links

Crossrefs

Cf. A344964, A344965, A344967, A344968.
Cf. A000796, A001620, A002117.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

Programs

  • Mathematica
    RealDigits[Pi^4/9 + 2*EulerGamma^2*Pi^2/3 + 4*EulerGamma*Zeta[3] + EulerGamma^4, 10, 100][[1]]

Formula

Equals Pi^4/9 + 2*gamma^2*Pi^2/3 + 4*gamma*zeta(3) + gamma^4, where gamma is Euler's constant (A001620).

A344967 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - 1), where x_k is the k-th zero of the digamma function.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 03 2021

Keywords

Comments

The zeros of the digamma function, i.e., the roots of psi(x) = 0 are x_0 = 1.461632... (A030169), the only positive root, x_1 = -0.504083... (A175472), etc.

Examples

			0.71349472210996816769933594441333563665531893958512...

Links

Crossrefs

Cf. A344964, A344965, A344966, A344968.
Cf. A000796, A001620, A072691.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

Programs

  • Mathematica
    RealDigits[Pi^2/(12*EulerGamma) + EulerGamma/2 - 1, 10, 100][[1]]

Formula

Equals Pi^2/(12*gamma) + gamma/2 - 1, where gamma is Euler's constant (A001620).

A344968 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - x_k), where x_k is the k-th zero of the digamma function.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jun 03 2021

Keywords

Comments

The zeros of the digamma function, i.e., the roots of psi(x) = 0 are x_0 = 1.461632... (A030169), the only positive root, x_1 = -0.504083... (A175472), etc.

Examples

			3.42698944421993633539867188882667127331063787917025...

Links

Crossrefs

Cf. A344964, A344965, A344966, A344967.
Cf. A001620, A013661.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

Programs

  • Mathematica
    RealDigits[Pi^2/(6*EulerGamma) + EulerGamma, 10, 100][[1]]

Formula

Equals Pi^2/(6*gamma) + gamma, where gamma is Euler's constant (A001620).
Showing 1-7 of 7 results.

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

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