

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
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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/(0.5frac(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

Table of n, a(n) for n=1..13.


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(exvalround(exval))<c) then c := (abs(exvalround(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

Cf. A079490, A137994, A137995.
Sequence in context: A116052 A141509 A054313 * A157190 A152657 A299172
Adjacent sequences: A080049 A080050 A080051 * A080053 A080054 A080055


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



