A060295 -id:A060295 - cp's OEIS frontend

A253214 Decimal expansion of log(640320^3)/sqrt(163), a Ramanujan constant producing 16 correct digits of Pi.

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

Offset: 1

Jean-François Alcover, Dec 29 2014

			3.141592653589793|0164958886531967237464..., where | shows the position from where the digits differ.

Jonathan M. Borwein, Aesthetics for the Working Mathematician
Eric Weisstein's MathWorld, Ramanujan Constant
Index entries for transcendental numbers

Cf. A000796, A060295.

RealDigits[Log[640320^3]/Sqrt[163], 10, 104] // First

log(640320^3)/sqrt(163) \\ Charles R Greathouse IV, Apr 20 2016

A076303 Engel expansion of exp(Pi * sqrt(163)) - 262537412640768743.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 19, 1169, 21384, 520409, 2559029, 2922819, 3228884, 6972029, 18244654, 24601850, 146539491, 620041946, 865572355, 1298955860, 3005000777, 5169423076, 6941400197, 9965578146, 26183561695, 39614218376

Offset: 1

Robert G. Wilson v, Mar 03 2003

nonn

262537412640768743.9999999999992500... is Ramanujan's constant which is extremely close to an integer. The Engel expansion of the fractional part begins with 40 terms 2.

OEIS wiki: Ramanujan's constant

Cf. A006784, A060295.

EngelExp[ A_, n_ ] := Join[ Array[ 1 &, Floor[ A ]], First@ Transpose @ NestList[ {Ceiling[ 1/Expand[ #[[ 1 ]] #[[ 2 ]] - 1 ]], Expand[ #[[ 1 ]] #[[ 2 ]] - 1]} &, {Ceiling[ 1/(A - Floor[A]) ], A - Floor[A]}, n - 1 ]]; EngelExp[E^(Pi*Sqrt[163]) - 262537412640768743, 52]

default(realprecision, 100000); r=exp(Pi*sqrt(163))-262537412640768743; for(i=1, 100, s=r*ceil(1/r)-1; print1(ceil(1/r), ", "); r=s); /* Georg Fischer, Nov 21 2020 */

More terms from Georg Fischer, Nov 21 2020

A102645 Decimal expansion of (Pi*sqrt(163))^e.

Original entry on oeis.org

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

Offset: 5

Gerald McGarvey, Feb 01 2005

The rounded value of this constant is 22807, a prime of the form p^2 + 6 where p is prime (cf. A079141), a balanced prime of order four (cf. A082079), a smallest prime larger than a square of an n-th prime, a largest prime == 7 mod 8 with class number 2n+1 (cf. A002147) and a prime p such that x^36 = 2 has no solution mod p, but x^12 = 2 has a solution mod p (cf. A059668).

			22806.99923855613927170389893443311151175881662508330399...

Cf. A060295.

RealDigits[(Pi*Sqrt[163])^E, 10, 111][[1]] (* Robert G. Wilson v, Feb 04 2005 *)

A111310 Decimal expansion of (e^(Pi*sqrt(163)) - 744)^(1/3).

Original entry on oeis.org

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

Offset: 6

Michael Trott (mtrott(AT)wolfram.com) and Robert G. Wilson v, Jul 15 2005

			Equals 640319.99999999999999999999999939031735231947012650283553902603366...
while (A060295)^(1/3) = 640320.0000000006048637350490160394717418188...

Michael Trott, The Mathematica Guide Book for Programming, Springer, 2004, page 8.

Cf. A060295.

RealDigits[(E^(Pi*Sqrt[163]) - 744)^(1/3), 10, 124][[1]]

PARI
```
(exp(Pi*sqrt(163))-744)^(1/3)
```

A262674 Decimal expansion of the real root of x^3 - 6x^2 + 4x - 2.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 27 2015

Algebraic integer of degree 3. - Charles R Greathouse IV, Apr 18 2016

			5.318628217750185659109680153318022467721919808836900260228...

Tito Piezas III, The 163 Dimensions of the Moonshine Functions, A Collection of Algebraic Identities.
Index entries for algebraic numbers, degree 3

Cf. A060295.

RealDigits[Root[#^3 - 6#^2 + 4# - 2&, 1], 10, 106] // First

solve(x=5, 6, x^3 - 6*x^2 + 4*x - 2) \\ Michel Marcus, Sep 27 2015

polrootsreal(x^3-6*x^2+4*x-2)[1] \\ Charles R Greathouse IV, Apr 18 2016

Equals (1/3)*(6 + (135 - 3*sqrt(489))^(1/3) + (3*(45 + sqrt(489)))^(1/3)).
Also equals exp(Pi*i/24)*eta(tau)/eta(2*tau), where eta is Dedekind's eta function and tau = (1 + sqrt(163) i) / 2.
Equals 2 + A160332. - R. J. Mathar, Sep 29 2015

A266296 Decimal expansion of a number close to 24, related to the Ramanujan number e^(Pi*sqrt(163)).

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Mar 13 2016

			24.00000000000000105127873861411207506351157289072577264955435...

Tito Piezas III The Ramanujan pages, see section 22.
Eric Weisstein's MathWorld, Almost Integer

Cf. A060295, A097340, A114609, A114610.

QP = QPochhammer; r4A[tau_] := With[{q = Exp[2 I Pi tau]}, (1/q) (QP[q^2]^2/(QP[q] QP[q^4]))^24]; RealDigits[r4A[(1/2) Sqrt[-163]] - Exp[Pi Sqrt[163]], 10, 105][[1]]
(* or: *)
x = Root[#^3 - 6#^2 + 4# - 2&, 1]; RealDigits[x^24 - Exp[Pi Sqrt[ 163]], 10, 105][[1]]

x^24 - e^(Pi*sqrt(163)), where x is the real root of x^3 - 6x^2 + 4x - 2.

cp's OEIS Frontend

A253214 Decimal expansion of log(640320^3)/sqrt(163), a Ramanujan constant producing 16 correct digits of Pi.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A076303 Engel expansion of exp(Pi * sqrt(163)) - 262537412640768743.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A102645 Decimal expansion of (Pi*sqrt(163))^e.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A111310 Decimal expansion of (e^(Pi*sqrt(163)) - 744)^(1/3).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

A262674 Decimal expansion of the real root of x^3 - 6x^2 + 4x - 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A266296 Decimal expansion of a number close to 24, related to the Ramanujan number e^(Pi*sqrt(163)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula