1,2

Robert G. Wilson v used Mathematica with a changing number of digits to accommodate 24 digits right of the decimal point.

At 12128 the difference from an integer is 0.000016103224605297330719...

The sequence of rounded reciprocals of the distances, b(n) = round(1/(.5-frac(Pi^a(n)-.5))) = round(1/abs(round(Pi^a(n))-Pi^a(n))), starts { 7, 8, 159, 190, 270, 2665, 10811, 26577, 62099, 70718, ... } - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 06 2008

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 58, p. 21, Ellipses, Paris 2008.

First term is 1 because this is just pi=3.14159... Second term is 2 because pi^2=9.869604... which is 0.13039... away from its nearest integer. pi^3=31.00627, hence third term is 3. pi^58 is 0.00527.. away from its nearest integer.

b := array(1..2000): Digits := 8000: c := 1: pos := 0: for n from 1 to 2000 do: exval := evalf(Pi^n): if (abs(exval-round(exval))<c) then c := (abs(exval-round(exval))): pos := pos+1: b[pos] := n: print(n):fi: od:

Used Maple with 8000 digits of precision and examined all n up to 2000.

a = 1; Do[d = Abs[ Round[Pi^n] - N[Pi^n, Ceiling[ Log[10, Pi^n] + 24]]]; If[d < a, Print[n]; a = d], {n, 1, 25000}]

$MaxExtraPrecision = 10^9; a = 1; Do[d = Abs[ Round[Pi^n] - N[Pi^n, Ceiling[ Log[10, Pi^n] + 24]]]; If[d < a, Print[n]; a = d], {n, 1, 10^5}] (Propper)

(PARI) f=0; for( i=1, 99999, abs(frac(Pi^i)-.5)>f | next; f=abs(frac(Pi^i)-.5); print1(i", ")) - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 06 2008

Cf. A079490, A137994, A137995.

Sequence in context: A000656 A116052 A054313 this_sequence A097961 A124083 A112098

Adjacent sequences: A080049 A080050 A080051 this_sequence A080053 A080054 A080055

nonn

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 22 2003

More terms from Carlos Alves (cjsalves(AT)gmail.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 23 2003

One more term from Ryan Propper (rpropper(AT)stanford.edu), Nov 13 2005