|
| |
|
|
A060295
|
|
Decimal expansion of e^(Pi*sqrt(163)).
|
|
26
|
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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 to expand 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 dividable 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. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science 2064/2001.
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
Tito Piezas III The Ramanujan pages, see section 05.
|
|
|
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=(x-d)*10; write("b060295.txt", n, " ", d)); } \\ Harry J. Smith, Jul 03 2009
|
|
|
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 (zevi_35711(AT)yahoo.com), Mar 24 2001
|
|
|
STATUS
|
approved
|
| |
|
|
