

A060295


Decimal expansion of exp(Pi*sqrt(163)).


27



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

18,1


COMMENTS

From Alexander R. Povolotsky, Jun 23 2009, Apr 04 2012: (Start)
One could observe that the last four of Class Number 1 expressions in T. Piezas "Ramanujan Pages" could be expressed as the following approximation:
exp(Pi*sqrt(19+24*n)) =~ (24*k)^3 + 31*24
which gives 4 (four) "almost integer" solutions:
1) n = 0, 19+24*0 = 19, k = 4;
2) n = 1, 19+24*1 = 43, k = 40;
3) n = 2, 19+24*2 = 67, k = 220;
4) n = 6, 19+24*6 = 163, k = 26680; this of course is the case for Ramanujan constant vs. its integer counterpart approximation. (End)
From Alexander R. Povolotsky, Oct 16 2010, Apr 04 2012: (Start)
Also if one expands the left part above to exp(Pi*sqrt(b(n)))
where b(n) = {19, 25, 43, 58, 67, 163, 232, ...}
then the expression (exp(Pi*sqrt(b(n))))/m (where m is either integer 1 or 8 )
yields values being very close to whole integer value:
Note, that the first differences of b(n) are all divisible by 3, giving after the division:
{2, 6, 5, 3, 32, 33, ...}. (End)


REFERENCES

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 106.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 179.
Dimitris Vathis, Letter to N. J. A. Sloane, Apr 22 1985.


LINKS

Harry J. Smith, Table of n, a(n) for n = 18..20000
J. Blanck, Exact real arithmetic systems: results of competition, pp. 389393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science 2064/2001.
Richard E. Borcherds, MegaFavNumbers 262537412680768000, video (2020)
Alex Clark and Brady Haran, 163 and Ramanujan Constant, Numberphile video (2012).
Tito Piezas III The Ramanujan pages, see section 05.
Simon Plouffe, exp(pi*sqrt(163)) to 5000 digits
Simon Plouffe, exp(Pi*sqrt(163)), the Ramanujan number,to a precision of 2000 digits
C. Radoux, A Formula of Ramanujan(Text in French)
C. Radoux, A Formula of Ramanujan(Continued) (Text in French)
Eric Weisstein's World of Mathematics, Ramanujan Constant
Index entries for transcendental numbers


FORMULA

exp(Pi*sqrt(163)) = A199743(6)^3 + 744  7.4992... * 10^13.  Charles R Greathouse IV, Jul 15 2020


EXAMPLE

The Ramanujan number = 262537412640768743.99999999999925007259719818568887935...


MATHEMATICA

RealDigits[N[E^(Pi*Sqrt[163]), 110]][[1]]


PROG

(PARI) default(realprecision, 20080); x=exp(Pi*sqrt(163))/10^17; for (n=18, 20000, d=floor(x); x=(xd)*10; write("b060295.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009
(MAGMA) R:= RealField(); Exp(Pi(R)*Sqrt(163)) // G. C. Greubel, Feb 15 2018


CROSSREFS

Cf. A058292, A019297, A093436, A102912, A181045, A181165, A181166.
Sequence in context: A220279 A221188 A220532 * A102912 A064850 A151853
Adjacent sequences: A060292 A060293 A060294 * A060296 A060297 A060298


KEYWORD

nonn,easy,cons


AUTHOR

Jason Earls, Mar 24 2001


STATUS

approved



