editing

approved

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

First term is 1 because this is just Pi = 3.14159....

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.

Pi^3 = 31.00627..., hence third term is 3.

Pi^58 is 0.00527... away from its nearest integer.

$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 *)

approved

editing

editing

approved

Value of n such that for any value of n, piPi^n is closer to its nearest integer than any value of piPi^k for 1 <= k < n.

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

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, Apr 06 2008

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

$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_ *)

a(11)-a(13) from _Jeremy Elson, _, Nov 13 2011

approved

editing

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

editing

approved

(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

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

approved

editing

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

(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

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

editing

approved