A166531 -id:A166531 - cp's OEIS frontend

A166532 Decimal expansion of A060295^6.

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

Offset: 105

Mark A. Thomas, Oct 16 2009

A large near-integer obtained by taking the Ramanujan constant e^(Pi*sqrt(163)) to the sixth power. The constants for even higher powers are in general no longer near integers.

			327451666639079200503292535866541250265248788274691526825971156\
747731856100971255480468836963064283775072.000097175254162592084120177\
65659310106524359922985819691442056333282681...

Henri Cohen, A Course in Computational Algebraic Number Theory, 3., corr. print., Springer-Verlag Berlin Heidelberg New York, 1996 pp. 383.

G. C. Greubel, Table of n, a(n) for n = 105..10000
Math Overflow, Questions
Eric Weisstein, Almost Integer, MathWorld
Wikipedia, Heegner number

Cf. A166528, A166529, A166530, A166531.

RealDigits[Exp[Pi Sqrt[163]]^6,10,120][[1]] (* Harvey P. Dale, Nov 27 2011 *)

exp(6*sqrt(163)*Pi) \\ Charles R Greathouse IV, Nov 05 2014

Equals exp(6*Pi*sqrt(163)) = A166528^3 = A166529^2.

Formula edited and connected to other powers by R. J. Mathar, Feb 27 2010

Minor edits by Vaclav Kotesovec, Jul 04 2014

A166528 Decimal expansion of A060295^2.

Original entry on oeis.org

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

Offset: 35

Mark A. Thomas, Oct 16 2009

Near-integer obtained by squaring Ramanujan's constant e^(Pi*sqrt(163)).

			exp(Pi*sqrt(163))^2 = 68925893036109279891085639286943768.00000000016373864420923460757232906257...

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag Berlin Heidelberg New-York, 1996 pp. 383.

Eric Weisstein's World of Mathematics, Almost Integer
Wikipedia, Heegner number
Math Overflow, Questions - _Mark A. Thomas_, Oct 02 2010

Cf. A166529, A166530, A166531, A166532.

RealDigits[Exp[Pi Sqrt[163]]^2,10,120][[1]] (* Harvey P. Dale, May 21 2013 *)

exp(2*Pi*sqrt(163)) \\ Charles R Greathouse IV, Nov 12 2014

Edited by N. J. A. Sloane, Oct 18 2009

Example corrected by Harvey P. Dale, May 21 2013

A166529 Decimal expansion of A060295^3.

Original entry on oeis.org

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

Offset: 53

Mark A. Thomas, Oct 16 2009

Near-integer obtained by cubing Ramanujan's constant e^(Pi*sqrt(163)).

			exp(3*Pi*sqrt(163)) =
18095625621654510801615355531263454706630064771074975 +
0.99999999012369367124132765224724197908973084944718563892074288285...

Henri Cohen, A Course in Computational Algebraic Number Theory, 3., corr. print., Springer-Verlag Berlin Heidelberg New York, 1996 p. 383.

Math Overflow, Questions [From _Mark A. Thomas_, Oct 02 2010]
Eric Weisstein's World of Mathematics, Almost Integer
Wikipedia, Heegner number

Cf. A166528, A166530, A166531, A166532, A060295.

RealDigits[Exp[Pi Sqrt[163]]^3,10,120][[1]] (* Harvey P. Dale, Jun 25 2022 *)

Edited by N. J. A. Sloane, Oct 17 2009

Previous Mathematica program replaced by Harvey P. Dale, Jun 25 2022

A166530 Decimal expansion of exp(4Pisqrt(163)) (or A060295^4).

Original entry on oeis.org

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

Offset: 70

Mark A. Thomas, Oct 16 2009

Near-integer obtained by taking Ramanujan's constant e^(Pi*sqrt(163)) to the fourth power.

			exp^(4*Pi*sqrt(163)) = 47507787308251777254639209489097266182144917180394713663187474063...

Henri Cohen, 'A Course in Computational Algebraic Number Theory', Springer-Verlag Berlin Heidelberg New-York 1996, p. 383.

MathOverflow, Why are powers of exp(Pi*sqrt(163)) almost integers?
Eric Weisstein's World of Mathematics, Almost Integer
Wikipedia, Heegner number

Cf. A166528, A166529, A166531, A166532.

RealDigits[Exp[Pi Sqrt[163]]^4,10,120][[1]]  (* Harvey P. Dale, Apr 20 2011 *)

exp(4*Pi*sqrt(163)) \\ Charles R Greathouse IV, Nov 12 2014

A181045 Decimal expansion of A060295/24.

Original entry on oeis.org

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

Offset: 17

Mark A. Thomas, Sep 30 2010

This real number is close to the prime number 10939058860032031. Also, the only (single) integer values placed in the denominator that will generate 'near-integers' from this relation are the divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (cf. A018253). A total of 64 'near-integers' can be obtained from generating powers (1-8) of A060295 and dividing each by one of the divisors of 24. Example: The last (64th) 'near-integer' is A060295^8 = 2.25698985492608864738884...99926422461218840012234... *10^139 (which is split by ... for brevity), the digits of which close to the decimal point are ...218840.012234... . While this does not quite look like a 'near-integer' this is where the pattern of 0's and 9's in the decimal tail cease in the case. See A166532.

			A060295/24 = 10939058860032030.999999999999968753024883257737036639... This is almost the prime 10939058860032031.

G. C. Greubel, Table of n, a(n) for n = 17..10000
Math Overflow, Questions [From _Mark A. Thomas_, Oct 02 2010]
M. A. Thomas, Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies, HAL preprint Id: hal-01232022, 2015.

Cf. A166528, A166529, A166530, A166531.

Magma
```
R:= RealField(); Exp(Pi*Sqrt(163))/24;
```

E^(Pi Sqrt[163])/24
RealDigits[Exp[Pi Sqrt[163]]/24, 10, 100][[1]] (* G. C. Greubel, Feb 14 2018 *)

exp(Pi*sqrt(163))/24 \\ G. C. Greubel, Feb 14 2018

Equals exp(Pi * sqrt(163))/24.

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A166532 Decimal expansion of A060295^6.

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A166528 Decimal expansion of A060295^2.

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A166529 Decimal expansion of A060295^3.

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A166530 Decimal expansion of exp(4*Pi*sqrt(163)) (or A060295^4).

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A181045 Decimal expansion of A060295/24.

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A166530 Decimal expansion of exp(4Pisqrt(163)) (or A060295^4).