

A111510


If n is even then a(n) is the nearest integer to Pi^n/Zeta(n), otherwise a(n) is the nearest integer to (Pi^n  n*e)/Zeta(n).


2



6, 19, 90, 282, 945, 2976, 9450, 29725, 93555, 294029, 924042, 2903286, 9121612, 28657229, 90030845, 282842357, 888579011, 2791558571, 8769948430, 27551618646, 86555983553, 271923674412, 854273468992, 2683779334264
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OFFSET

2,1


COMMENTS

Lim_{n>inf.} i_n/i_(n1) approaches Pi. e.g. 2791558571/8885799011=~3.141598593...
See A108925. Analytical Pi (for n>=4 but here n>10^6 say),(n=1 2 3...n). Take n straight lines monotonically increasing in length by one and join them end to end; the last to the first. When the enclosed area is at its maximum every vertex will lie on the circumference of a circle the diameter of which divided into Triangular(n) equals Pi.
There is an interesting benchmark when n=8. The radius calculated using Pi equals 5.7296...; one tenth of the number of degrees in a radian. The radius when plotted as a drawing is very near to six and, tentatively, this could be ten times a constant near to point six.
It appears that a(2n1) taken when rounded down (rather than to the nearest integer) is equal to A100594(n).  Terry D. Grant, May 28 2017


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..1000


EXAMPLE

a(n) = d where d is the integer divisor of Pi^n for even n and (Pi^n)ne for odd n having a solution closest to Zeta(n).
a(2) = 6 then (Pi^2)/6 = Zeta(2); a(3)=19, (Pi^33e)/19 approx = Zeta(3); a(4)=90, (Pi^4)/90 = Zeta(4); and the only special case the author has found where ((Pi^4)4e)/80 approx = Zeta(4).


MATHEMATICA

f[n_] := Round@If[EvenQ@n, Pi^n/Zeta@n, (Pi^n  n*E)/Zeta@n]; Table[ f@n, {n, 2, 26}] (* Robert G. Wilson v, Nov 18 2005 *)


CROSSREFS

Cf. A100594, A108925.
Sequence in context: A026545 A041937 A279512 * A151277 A192368 A323686
Adjacent sequences: A111507 A111508 A111509 * A111511 A111512 A111513


KEYWORD

nonn


AUTHOR

Marco Matosic, Nov 16 2005


EXTENSIONS

Corrected and extended by Robert G. Wilson v, Nov 18 2005
Corrections from Marco Matosic, Mar 27 2006
Definition clarified by Omar E. Pol, Jan 02 2009


STATUS

approved



