A102302 Largest number < n/2 coprime to n.

3, 3, 4, 3, 5, 5, 6, 5, 7, 7, 8, 7, 9, 9, 10, 9, 11, 11, 12, 11, 13, 13, 14, 13, 15, 15, 16, 15, 17, 17, 18, 17, 19, 19, 20, 19, 21, 21, 22, 21, 23, 23, 24, 23, 25, 25, 26, 25, 27, 27, 28, 27, 29, 29, 30, 29, 31, 31, 32, 31, 33, 33, 34, 33, 35, 35, 36, 35, 37, 37, 38, 37, 39, 39

Offset: 7

Author

Hugo Pfoertner, Jan 23 2005

Keywords

• easy
• nonn

Comments

The densest possible star-shaped regular n-gon is formed by connecting with straight lines every a(n)-th point out of n regularly spaced points lying on a circumference.

For a given n there are A055684(n) different star-shaped regular polygons. The minimum skip increment for connecting points on the circumference is given by A053669(n), the maximum skip increment is given by a(n). There are no star-shaped polygons for n=3,4,6 and unique star-shaped polygons for n=5,8,10 and 12, for which a(n) = A053669(n).

Links

• Colin Barker, Table of n, a(n) for n = 7..1000
• Jay Kappraff, Gary W. Adamson, Polygons and Chaos, BRIDGES Mathematical Connections in Art, Music, and Science, 2001.
• Hugo Pfoertner, Star-shaped regular polygons.
• Hugo Pfoertner, (Star-Shaped-) Polygons with Maximal Density.
• Eric Weisstein's World of Mathematics, Star Polygon.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A053669 (least number coprime to n), A055684 (number of different n-pointed stars).

Programs

Mathematica

lnc[n_]:=Module[{k=Floor[n/2]},While[!CoprimeQ[n,k],k--];k]; Array[ lnc,90,7] (* Harvey P. Dale, May 15 2021 *)

PARI

Vec(-x^7*(x^4+x^3-x^2-3)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 21 2015

Formula

a(4*k-1) = a(4*k) = a(4*k+2) = 2*k-1; a(4*k+1) = 2*k.

a(n) = (1/2) (n - (I^n + (-I)^n)/2 - (-1)^n + 4). - Ralf Stephan, May 17 2007

a(n) = a(n-1)+a(n-4)-a(n-5) for n>11. - Colin Barker, Feb 21 2015

G.f.: -x^7*(x^4+x^3-x^2-3) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 21 2015