A060295 -id:A060295 - 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

A166531 Decimal expansion of A060295^5.

Original entry on oeis.org

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

Offset: 88

Mark A. Thomas, Oct 16 2009

A large near-integer obtained by taking the Ramanujan constant e^(Pi*sqrt(163)) to the fifth power.

			Equals 1247257156019637304856107520018074552566824585862995272173368815\
794085495792299621093743.99999365418746...

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

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

Cf. A166528, A166529, A166530, A166532.

RealDigits[Exp[Pi Sqrt[163]]^5,10,120][[1]] (* Harvey P. Dale, Aug 24 2025 *)

Equals exp(5*Pi*sqrt(163)) = A166529*A166528.

Keyword:cons added by R. J. Mathar, Feb 27 2010

Previous Mathematica program replaced by Harvey P. Dale, Aug 24 2025

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.

A019297 Integers k such that abs(e^(Pi*sqrt(n)) - k) < 0.01 for some n.

Original entry on oeis.org

-1, 1, 2198, 422151, 614552, 2508952, 6635624, 199148648, 884736744, 24591257752, 30197683487, 147197952744, 545518122090, 70292286279654, 39660184000219160, 45116546012289600, 262537412640768744

Offset: 0

Roy Williams Clickery (roy(AT)ccsf.caltech.edu)

sign

Old name of sequence was "Integers that are very close to values of exp(Pi*sqrt(n))", which left "very close" undefined. Robert G. Wilson v resolved this problem on Feb 28 2006 with the comment that "'Very close' means to within 0.01." - Jon E. Schoenfield, Mar 21 2015

			e^(Pi*sqrt(163)) = 262537412640768743.99999999999925007259719818568887935385...

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 179.

Cf. A019296, A058292, A060295.

f[n_] := Block[{e = Exp[Pi*Sqrt[n]]}, Abs[e - Round[e]]]; Round @ Exp[Pi*Sqrt @ Select[Range[ -1, 200], f @ # < 10^(-2) &]] (* Robert G. Wilson v, Feb 28 2006 *)

New name, based on Feb 28 2006 comment from Robert G. Wilson v, from Jon E. Schoenfield, Mar 21 2015

Typos in Mma program corrected by Giovanni Resta, Jun 12 2016

A102912 Decimal expansion of a close approximation to the Ramanujan constant.

Original entry on oeis.org

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

Offset: 18

Eric W. Weisstein, Jan 17 2005

First differs from Ramanujan's constant (A060295) at a(33). - Omar E. Pol, Jun 26 2012

Kontsevich & Zagier give also exp(3*log(640320)) = 2.62537412640768000... as a close approximation to the Ramanujan constant. - Jean-François Alcover, Jun 22 2015

			262537412640768743.999999999999251123875936799800954417367910227716...

G. C. Greubel, Table of n, a(n) for n = 18..10000
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22
Eric Weisstein's World of Mathematics, Ramanujan Constant

Cf. A060295.

RealDigits[ Root[ #^3 - 6#^2 + 4# - 2 &, 1]^24 - 24, 10, 111][[1]]

Equals: Real root of x^3 - 6*x^2 + 4*x - 2 = 0, being x_{real} = (6 + (3*(45 + sqrt(489)))^(1/3) + (3*(45 - sqrt(489)))^(1/3))/3 = 5.31863, evaluated as (x_{real})^24 - 24. - G. C. Greubel, Feb 15 2018

A058292 Continued fraction for e^(Pi*sqrt(163)).

Original entry on oeis.org

262537412640768743, 1, 1333462407511, 1, 8, 1, 1, 5, 1, 4, 1, 7, 1, 1, 1, 9, 1, 1, 2, 12, 4, 1, 15, 4, 299, 3, 5, 1, 4, 5, 5, 1, 28, 3, 1, 9, 4, 1, 6, 1, 1, 1, 1, 1, 1, 51, 11, 5, 3, 2, 1, 1, 1, 1, 2, 1, 5, 1, 9, 1, 2, 14, 1, 82, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

The real number e^(pi*sqrt(163)) ~ a(0)+1-1/a(2) (cf also the Example section) is called Ramanujan's constant: See the main entry A060295 for further information. - M. F. Hasler, Jan 26 2014

			e^(Pi*Sqrt(163)) = 262537412640768743.99999999999925007259719818568887935385...

Flajolet, Philippe, and Brigitte Vallée. "Continued fractions, comparison algorithms, and fine structure constants." Constructive, Experimental, and Nonlinear Analysis 27 (2000): 53-82. See Fig. 3.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 179.

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A060295, A019297.

ContinuedFraction[ E^(Pi*Sqrt[163]), 100 ]

default(realprecision,99);contfrac(exp(Pi*sqrt(163))) \\ With standard precision (38 digits), contfrac() returns only [a(0)+1]. - M. F. Hasler, Jan 26 2014

A160514 A nonsense sequence.

Original entry on oeis.org

256, 9, 25, 49, 119663008743245277588690345984961

Offset: 1

Mark A. Thomas, May 16 2009

Previous name was: The sequence is: 2^8, 3^2, 5^2, 7^2, 10939058860032031^2 where 10939058860032031 is prime and where its order is an integer 337736875876935471466319632507953926400 and is equal to ((640320)^3 + 744)^2 * 70^2 where (640320)^3 + 744 is an integer value which is nearly Ramanujan's constant and 1^2 + 2^2 + 3^2 + ... + 22^2 + 23^2 + 24^2 = 70^2 which is related to the norm vector 0 used in construction of the Leech lattice.

The prime number 10939058860032031 = 2^15, 3^2, 5^3, 23^3, 29^3 + 31 Ramanujan's constant: e^(Pi*sqrt(163))= 640320^3 + 743.99999999999925 = A060295.

Mark A. Thomas, Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies, HAL preprint Id: hal-01232022, 2015.

Power[#1, #2] & @@@ FactorInteger[(640320^3 + 744)^2*70^2] (* Michael De Vlieger, Dec 19 2015 *)

(640320^3 + 744)^2 * 70^2 = A160515.

Partially edited by R. J. Mathar, May 30 2009

cp's OEIS Frontend

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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A166531 Decimal expansion of A060295^5.

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A181045 Decimal expansion of A060295/24.

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A019297 Integers k such that abs(e^(Pi*sqrt(n)) - k) < 0.01 for some n.

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