A060295 -id:A060295 - cp's OEIS frontend

A093436 Decimal expansion of exp(Pi*sqrt(67)).

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

Offset: 12

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), May 13 2004

			147197952743.99999866245422450682926131257862850818331250381671263337...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 225-226.

G. C. Greubel, Table of n, a(n) for n = 12..10000

Cf. A060295.

R:= RealField(); Exp(Pi(R)*Sqrt(67)); // G. C. Greubel, Feb 13 2018

RealDigits[E^(Pi*Sqrt[67]), 10, 111][[1]] (* Robert G. Wilson v, May 24 2004 *)

PARI
```
exp(Pi*sqrt(67))
```

More terms from Pab Ter (pabrlos(AT)yahoo.com) and Robert G. Wilson v, May 23 2004

A190575 Decimal expansion of exp(Pi*sqrt(163)/3).

Original entry on oeis.org

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

Offset: 6

N. J. A. Sloane, May 12 2011

			640320.00000000060486373504901603947174181881853947577148576...

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 128.

Cf. A190574.

RealDigits[Exp[(Pi Sqrt[163])/3],10,120][[1]] (* Harvey P. Dale, Jan 21 2019 *)

default(realprecision,1000);
first(n)=digits(floor(10^(n-6)*exp(Pi*sqrt(163)/3))) \\ Edward Jiang, Sep 07 2014

Equals cube root of A060295. - Michel Marcus, Sep 08 2014

A212131 Decimal expansion of k such that e^(ksqrt(163)) = round(e^(Pisqrt(163))).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 25 2012

Decimal expansion of log(262537412640768744)/sqrt(163).
First differs from A000796 at a(32).
Note that 262537412640768744 = 24*10939058860032031 = 2^3 * 3 * 10939058860032031, is the nearest integer to the value of Ramanujan's constant e^(Pi*sqrt(163)) = 262537412640768743.999999999999250... = A060295.

			3.14159265358979323846264338327972661934754988... (very close to Pi).

Eric Weisstein's World of Mathematics, Ramanujan's constant

Cf. A000796, A003173, A019297, A060295, A080283, A102912, A160514, A160515, A181045.

RealDigits[Log[Round[E^(Pi Sqrt[163])]]/Sqrt[163], 10, 105][[1]] (* Bruno Berselli, Jun 26 2012 *)

k = log(round(e^(Pi*sqrt(163))))/sqrt(163).

More terms from Alois P. Heinz, Jun 25 2012

A100378 Decimal expansion of exp(Pi*sqrt(89/3)).

Original entry on oeis.org

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

Offset: 8

Gerald McGarvey, Dec 29 2004

Let q = this constant, then q = 300^3 + Sum_{k=0...infinity} (-1)^i*A030197(i)/q^i where A030197 is the McKay-Thompson series of class 3A for Monster, expansion of Hauptmodul for X_0^{+}(3) and expansion of (h+27)^2/h, where h = (eta(q)/eta(q^3))^12.

			27000041.999971000057006717059989970669906272703208212...

G. C. Greubel, Table of n, a(n) for n = 8..10000
Titus Piezas III, Pi Formulas

Cf. A030197, A007243, A045480, A060295.

R:= RealField(); Exp(Pi(R)*Sqrt(89/3)); // G. C. Greubel, Feb 14 2018

RealDigits[Exp[Pi*Sqrt[89/3]], 10, 100][[1]] (* G. C. Greubel, Feb 14 2018 *)

exp(Pi*sqrt(89/3)) \\ G. C. Greubel, Feb 14 2018

A100379 Decimal expansion of exp(Pi*sqrt(29/2)).

Original entry on oeis.org

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

Offset: 6

Gerald McGarvey, Dec 29 2004

Let q = this constant, then q = 396^2 - Sum_{k>=0} A007247(k)/q^(2*k - 1) where A007247 is the McKay-Thompson series of class 4B for Monster.

			156815.99966840118263523959861196710809095351232844202...

G. C. Greubel, Table of n, a(n) for n = 6..10000
Titus Piezas III The Ramanujan Pages

Cf. A060295.

R:= RealField(); Exp(Pi(R)*Sqrt(29/2)); // G. C. Greubel, Feb 14 2018

RealDigits[Exp[Pi Sqrt[29/2]],10,120][[1]] (* Harvey P. Dale, Jul 19 2011 *)

exp(Pi*sqrt(29/2)) \\ G. C. Greubel, Feb 14 2018

A100811 Decimal expansion of exp(Pi*sqrt(29/8)).

Original entry on oeis.org

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

Offset: 3

Gerald McGarvey, Jan 05 2005

Let q = this constant, then q = 396 - Sum_{k>=1} A052241(k)/q^(4k-1) where A052241 is the McKay-Thompson series of class 8C for Monster.

			395.9999995813146243181087813157584866388138244764778475...

G. C. Greubel, Table of n, a(n) for n = 3..10000
Titus Piezas III The Ramanujan Pages

Cf. A052241, A060295.

R:= RealField(); Exp(Pi(R)*Sqrt(29/8)); // G. C. Greubel, Feb 13 2018

RealDigits[Exp[Pi Sqrt[29/8]],10,120][[1]] (* Harvey P. Dale, Feb 08 2015 *)

exp(Pi*sqrt(29/8)) \\ G. C. Greubel, Feb 13 2018

A210963 Decimal expansion of sqrt(163).

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Jun 26 2012

Also decimal expansion of log(R_c)/Pi, where R_c is Ramanujan's constant: 262537412640768743.999999999999250... = A060295.

			163^(1/2) = 12.76714533480370466171095200978...

Eric Weisstein's World of Mathematics, Ramanujan's constant

Cf. A000796, A060295, A102912, A210965, A212131.

RealDigits[Sqrt[163], 10, 105][[1]] (* T. D. Noe, Jun 27 2012 *)

PARI
```
sqrt(163) \\ Michel Marcus, Mar 26 2017
```

A210965 Decimal expansion of k such that e^(Pik) = round(e^(Pisqrt(163))).

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Jun 26 2012

Decimal expansion of k = log(262537412640768744)/Pi.
Note that 262537412640768744 = 24*10939058860032031 = 2^3 * 3 * 10939058860032031, is the nearest integer to the value of Ramanujan's constant e^(Pi*sqrt(163)) = A060295.
By construction, this constant here is very close to sqrt(163) = A210963.

			12.767145334803704661710952...

Eric Weisstein's World of Mathematics, Ramanujan constant

Cf. A000796, A060295, A102912, A212131.

RealDigits[Log[262537412640768744]/Pi,10,120][[1]] (* Harvey P. Dale, Nov 12 2017 *)

Equals log(round(e^(Pi*sqrt(163))))/Pi.

A226120 Decimal expansion of Sum_{n>=1} n^3/(exp(2Pin/7)-1).

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, May 27 2013

An almost-integer discovered by Simon Plouffe. The computed sum equals 10 within 15 digits.

			10.00000000000000019016176788866267558435930585544533348025489784340610994387...

Simon Plouffe, Simon Plouffe Home Page
Eric Weisstein's MathWorld, Almost Integer

Cf. A060295 (famous almost-integer: Ramanujan's constant), A226121 (another surprising almost-integer by Simon Plouffe), A007775, A089034.

NSum[n^3/(Exp[2*Pi*n/7] - 1), {n, 1, Infinity}, NSumTerms -> 220,    WorkingPrecision -> 100] // RealDigits[#, 10, 100] & // First

Offset corrected by Rick L. Shepherd, Jan 01 2014

A226121 Decimal expansion of sum_{n=1..infinity} n^3/(exp(2Pin/13)-1).

Original entry on oeis.org

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

Offset: 3

Jean-François Alcover, May 27 2013

An almost-integer discovered by Simon Plouffe. The computed sum equals 119 within 31 digits after the decimal point.

			119.0000000000000000000000000000000959374585102554733558858491339738415043391...

Simon Plouffe, Simon Plouffe Home Page
Eric Weisstein's MathWorld, Almost Integer

Cf. A060295 (a famous almost-integer: Ramanujan's constant), A226120 (another surprising almost-integer by Simon Plouffe).

NSum[n^3/(Exp[2*Pi*n/13] - 1), {n, 1, Infinity}, NSumTerms -> 500,    WorkingPrecision -> 100] // RealDigits[#, 10, 100] & // First

Offset corrected by Rick L. Shepherd, Jan 01 2014

cp's OEIS Frontend

A093436 Decimal expansion of exp(Pi*sqrt(67)).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A190575 Decimal expansion of exp(Pi*sqrt(163)/3).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

A212131 Decimal expansion of k such that e^(k*sqrt(163)) = round(e^(Pi*sqrt(163))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A100378 Decimal expansion of exp(Pi*sqrt(89/3)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A100379 Decimal expansion of exp(Pi*sqrt(29/2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A100811 Decimal expansion of exp(Pi*sqrt(29/8)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A210963 Decimal expansion of sqrt(163).

Views

Author

Keywords

Comments

Examples

Links

A212131 Decimal expansion of k such that e^(ksqrt(163)) = round(e^(Pisqrt(163))).

A210965 Decimal expansion of k such that e^(Pik) = round(e^(Pisqrt(163))).

A226120 Decimal expansion of Sum_{n>=1} n^3/(exp(2Pin/7)-1).

A226121 Decimal expansion of sum_{n=1..infinity} n^3/(exp(2Pin/13)-1).