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A080052 Value of n such that for any value of n, Pi^n is closer to its nearest integer than any value of Pi^k for 1 <= k < n. 10
1, 2, 3, 58, 81, 157, 1030, 5269, 12128, 65875, 114791, 118885, 151710 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Robert G. Wilson v used Mathematica with a changing number of digits to accommodate 24 digits to the 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/(0.5-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, Apr 06 2008
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 58, p. 21, Ellipses, Paris 2008.
LINKS
EXAMPLE
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.
MAPLE
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.
MATHEMATICA
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}] (* Ryan Propper, Nov 13 2005 *)
PROG
(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, Apr 06 2008
CROSSREFS
Sequence in context: A116052 A141509 A054313 * A157190 A152657 A299172
KEYWORD
nonn
AUTHOR
Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 22 2003
EXTENSIONS
More terms from Carlos Alves and Robert G. Wilson v, Jan 23 2003
One more term from Ryan Propper, Nov 13 2005
a(11)-a(13) from Jeremy Elson, Nov 13 2011
STATUS
approved

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Last modified April 19 21:09 EDT 2024. Contains 371798 sequences. (Running on oeis4.)