A100594 -id:A100594 - cp's OEIS frontend

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).

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

Offset: 2

Marco Matosic, Nov 16 2005

nonn

Lim_{n->inf.} i_n/i_(n-1) 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(2n-1) taken when rounded down (rather than to the nearest integer) is equal to A100594(n). - Terry D. Grant, May 28 2017

			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^3-3e)/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).

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

Cf. A100594, A108925.

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 *)

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

A309946 a(n) = floor(Pi^n/Zeta(n)).

Original entry on oeis.org

0, 6, 25, 90, 295, 945, 2995, 9450, 29749, 93555, 294058, 924041, 2903320, 9121612, 28657269, 90030844, 282842403, 888579011, 2791558622, 8769948429, 27551618702, 86555983552, 271923674474, 854273468992, 2683779334331, 8431341566236, 26487840921750, 83214006759229, 261424512797515

Offset: 1

Seiichi Manyama, Aug 24 2019

nonn

			Pi^12/Zeta(12) = 638512875/691 = 924041.78... So a(12) = 924041.

Index entries for zeta function.

Decimal expansion of Pi^k/Zeta(k): A308637 (k = 3), A309926 (k = 5), A309927 (k = 7), A309928 (k = 9), A309929 (k = 11).

Cf. A001672 (floor(Pi^n)), A002432, A046988, A100594.

Table[Floor[Pi^n/Zeta[n]], {n, 20}] (* Alonso del Arte, Aug 24 2019 *)

{a(n) = if(n==1, 0, n==4, 90, floor(Pi^n/zeta(n)))}

a(2*n) = A100594(n).

cp's OEIS Frontend

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).

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