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A102302
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Largest number < n/2 coprime to n.
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4
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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
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OFFSET
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7,1
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COMMENTS
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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).
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LINKS
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Jay Kappraff, Gary W. Adamson, Polygons and Chaos, BRIDGES Mathematical Connections in Art, Music, and Science, 2001.
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FORMULA
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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
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MATHEMATICA
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lnc[n_]:=Module[{k=Floor[n/2]}, While[!CoprimeQ[n, k], k--]; k]; Array[ lnc, 90, 7] (* Harvey P. Dale, May 15 2021 *)
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PROG
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(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
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CROSSREFS
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Cf. A053669 (least number coprime to n), A055684 (number of different n-pointed stars).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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