

A127031


Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n)  floor(f(n)) < 1/10^6.


11



652, 2608, 880111, 2720885, 4089051, 4619054, 5046630, 5409046, 5433402, 5603556, 5645558, 7278138, 7466589, 10037029, 10730786, 10823358, 11540978, 11860073, 12898258, 14554227, 15107659, 15602035, 15896143, 17070573, 18204473, 19252185, 19425342, 19556500
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OFFSET

1,1


COMMENTS

a(212)=195246501 is the smallest integer such that the fractional part of e^(Pi*sqrt(n)) begins with exactly 8 zeros.  Anthony Canu, Oct 11 2017
The probability of finding two real numbers of the form e ^ (Pi * sqrt (n)) such that their fractional parts begins with exactly 8 zeros for n belonging to [195000000; 205000000] is less than 5/1000 and yet e^(Pi*sqrt(204990857)) begins also with exactly 8 zeros.  Anthony Canu, Oct 29 2017


LINKS

Anthony Canu, Table of n, a(n) for n = 1..217


EXAMPLE

5 is not in the sequence since exp(Pi*sqrt(5)) = 1124.186... has fractional part 0.186... which is greater than 1/10^6. But exp(Pi*sqrt(652)) has fractional part 0.0000000001637... which is less than 1/10^6, so 652 is in the sequence.  Michael B. Porter, Aug 24 2016


MATHEMATICA

$MaxExtraPrecision = 1000; a = {}; Do[If[((Exp[Pi Sqrt[x]]  Floor[Exp[Pi Sqrt[x]]]) > 0) && ((Exp[Pi Sqrt[x]]  Floor[Exp[Pi Sqrt[x]]]) < 10^(6)), AppendTo[a, x]], {x, 1, 100000}]; a


PROG

(PARI) search(a, b)=my(t, prec=default(realprecision), nprec=round(Pi*sqrt(b)/log(10)+20)); default(realprecision, nprec); for(n=floor(a), b, t=exp(Pi*sqrt(n)); if(tfloor(t)<.000001, print(n))); default(realprecision, prec) \\ Charles R Greathouse IV, Jul 28 2009


CROSSREFS

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127027, A127028, A127029.
Sequence in context: A002232 A127029 A127030 * A204511 A204506 A282558
Adjacent sequences: A127028 A127029 A127030 * A127032 A127033 A127034


KEYWORD

nonn


AUTHOR

Artur Jasinski, Jan 03 2007


EXTENSIONS

a(3)a(14) from Charles R Greathouse IV, Jul 28 2009
a(15)a(18) from Anthony Canu, Aug 24 2016
a(19) from Anthony Canu, Aug 31 2016
a(20)a(28) from Anthony Canu, Mar 03 2017


STATUS

approved



