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.

Previous Showing 21-28 of 28 results.

A347074 Decimal expansion of sqrt(1 + sqrt(2 + sqrt(3 + sqrt(5 + sqrt(7 + ... + sqrt(prime(k) + ...)))))).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Aug 15 2021

Keywords

Examples

			1.761703010254536989908177136637182102146955...

Links

Crossrefs

Cf. A008578, A072449, A105546, A239349.

A352495 Decimal expansion of the pearl of the Riemann zeta function.

Original entry on oeis.org

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

Views

Author

Eduard Roure Perdices, Mar 18 2022

Keywords

Comments

Let Z be the Riemann zeta function, and consider its sequence of nontrivial zeros with nonnegative imaginary part, {r(m)}, so that for every m >= 1, Z(r(m)) = 0, 0 <= Re(r(m)) <= 1, and 0 <= Im(r(m)), and for every k > m, Im(r(m)) < Im(r(k)), or Im(r(m)) = Im(r(k)) and Re(r(m)) < Re(r(k)).
Let i be the imaginary unit, and define the sequence {b(m)} as follows: b(1) = Z((r(1)-1/2)/i), b(2) = Z((r(1)-1/2)/i + Z((r(2)-1/2)/i)), b(3) = Z((r(1)-1/2)/i + Z((r(2)-1/2)/i + Z((r(3)-1/2)/i))), and so on. If this sequence converges, we call its limit the pearl of Z.
Suppose that the Riemann Hypothesis is true. Then the sequence {b(m)} is real. On the interval [2,oo), Z is decreasing, positive, and bounded above by 2, so {b(2*m-1)} is decreasing and bounded below by 0, and hence, it converges to a real value, say A. Moreover, {b(2*m)} is increasing and b(2*m) <= b(2*m+1), and by repeated application of the mean value theorem, b(2*m+1) - b(2*m) <= Z(Im(r(2*m+1))) * |Z'(Im(r(1)))|^(2*m) <= 2*(4/100000)^(2*m), so {b(2*m)} also converges to A, and {a(n)} is the decimal expansion of this value.
We don't know if the existence of a real pearl of Z implies the Riemann Hypothesis.
More generally, the definition of pearl works for Dirichlet L-functions, giving rise to analogous constants, not necessarily real.

Examples

			1.00002785763306644073021509185736217782971009140533304787973192845864...

Links

Crossrefs

Cf. A072449, A099874, A099876, A099877, A105546, A105817, A151558, A239349, A277313, A278812.

Programs

  • Mathematica
    RealDigits[Re[res = Fold[Zeta[#1 + #2] &, 0, Reverse[(ZetaZero[Range[10]] - 1/2)/I]]], 10, 100][[1]]

A365747 Decimal expansion of Trinv(1 + Trinv(2 + Trinv(3 + Trinv(4 + ... )))) where Trinv(n) = (sqrt(8*n+1)-1)/2.

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, Sep 17 2023

Keywords

Comments

Trinv(n) = (sqrt(8*n+1)-1)/2 is the inverse of A000217.

Examples

			2.2814374140534244152717727285916507607333884526610...

Crossrefs

Cf. A072449 (analog for square root), A099874 (analog for cube root).
Cf. A000217 (triangular numbers), A003056.

Programs

  • Mathematica
    TriangleRoot[n_] =(-1 + Sqrt[1 + 8 n])/2; RealDigits[ Fold[ TriangleRoot[ #1 + #2] &, 0, Reverse[ Range[200]]], 10,111][[1]]

A285814 Decimal expansion of the limit of the nested logarithm log(1+2*log(1+3*log(1+4*log(...)))).

Original entry on oeis.org

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

Views

Author

Alex Klotz and Robert G. Wilson v, Apr 27 2017

Keywords

Comments

No closed form expression is known. Probably transcendental but this is unproved.

Examples

			1.6630272645956895211326676626846841888930929806388191033183243319627019656...

Links

Crossrefs

Cf. A072449, A276521, A276538, A276539, A277313, A278812, A283749.

Programs

  • Mathematica
    bgn = 147; RealDigits[ Fold[ N[1 + #2*Log@#1, 200] &, bgn +1, Reverse@Range@bgn] -1, 10, 111][[1]]

A296040 Decimal expansion of sqrt(1^1 + sqrt(2^2 + sqrt(3^3 + sqrt(4^4 + sqrt(5^5 + ...))))).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Examples

			2.0661766864240510842686019157724311005106721685821046...

Links

Crossrefs

Cf. A000312, A072449, A099876, A099879, A105546, A105817, A239349, A257574, A257575, A257576, A257577, A257578.

A296042 Decimal expansion of sqrt(1 + 2*sqrt(2 + 3*sqrt(3 + 4*sqrt(4 + 5*sqrt(5 + 6*sqrt(6 + ...)))))).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Comments

Decimal expansion of sqrt(1 + sqrt(8 + sqrt(432 + sqrt(1327104 + ... + sqrt(k*A030450(k) + ...))))).

Examples

			3.0833551418306944580511425800881719306014784933...

Links

Crossrefs

Cf. A030450, A072449, A099876, A099879, A105546, A105817, A239349, A254689, A257574, A257575, A257576, A257577, A257578.

A347073 Decimal expansion of sqrt(1*2 + sqrt(2*3 + sqrt(3*4 + sqrt(4*5 + sqrt(5*6 + ...))))).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Aug 15 2021

Keywords

Examples

			2.276781083523092607713497171325417324079...

Links

Crossrefs

Cf. A002378, A072449, A099876, A099879, A296041.

A373009 Decimal expansion of PentInv(1 + PentInv(2 + PentInv(3 + PentInv(4 + ... )))) where PentInv(n) = (1 + sqrt(1 + 24*n))/6.

Original entry on oeis.org

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

Views

Author

Kelvin Voskuijl, May 22 2024

Keywords

Comments

The inverse of the pentagonal numbers is defined here as (1 + sqrt(1 + 24*n))/6.

Examples

			1.54715758729787666125880794008651848362064131346850513...

Crossrefs

Cf. A365747 (triangular inverse), A072449 (square root), A099874 (cubes).
Cf. A000326 (pentagonal numbers).
Previous Showing 21-28 of 28 results.

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