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 A127031 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^6. 11

%I

%S 652,2608,880111,2720885,4089051,4619054,5046630,5409046,5433402,

%T 5603556,5645558,7278138,7466589,10037029,10730786,10823358,11540978,

%U 11860073,12898258,14554227,15107659,15602035,15896143,17070573,18204473,19252185,19425342,19556500

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

%C 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

%C The probability of finding two real numbers of the form e^(Pi*sqrt(n)) whose fractional parts begin with exactly 8 zeros for n in the interval [195000000, 205000000] is less than 5/1000, yet e^(Pi*sqrt(204990857)) also begins with exactly 8 zeros. - _Anthony Canu_, Oct 29 2017

%H Anthony Canu, <a href="/A127031/b127031.txt">Table of n, a(n) for n = 1..217</a>

%e 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

%t \$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

%o (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(t-floor(t)<.000001, print(n))); default(realprecision,prec) \\ _Charles R Greathouse IV_, Jul 28 2009

%Y Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127027, A127028, A127029.

%K nonn,changed

%O 1,1

%A _Artur Jasinski_, Jan 03 2007

%E a(3)-a(14) from _Charles R Greathouse IV_, Jul 28 2009

%E a(15)-a(18) from _Anthony Canu_, Aug 24 2016

%E a(19) from _Anthony Canu_, Aug 31 2016

%E a(20)-a(28) from _Anthony Canu_, Mar 03 2017

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)