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%I M5144 #63 Jan 01 2023 04:27:39
%S 24,54,216,648,2376,8100,29232,104544,381672,1397070,5163480,19170432,
%T 71587080,268423200,1010595960,3817704744,14467313448,54975424194,
%U 209430985176,799644248064,3059511345864,11728121930100,45035998958016,173215362539520,667199954727936
%N Number of n-gons in cubic curve.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A005782/b005782.txt">Table of n, a(n) for n = 3..1000</a>
%H M. Picquet, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k433703n/f35">Applications de la représentation des courbes du troisième degré</a>, Journal de l'École Polytechnique, Paris, 35 (1884), pp. 31-100. See p. 54.
%F When n is a prime Picquet gives a simple formula for a(n) - see A182589.
%F His formula for composite n is more complicated:
%F "Pour calculer le nombre propre des sommets des polygones de n côtés, on formera tous les diviseurs a de n complémentaires des diviseurs du même nombre qui n'admettent leurs facteurs premiers qu'à la première puissance, et si le nombre de ces facteurs est pair, on ajoutera à phi(n) ou chi(n), suivant que n est impair ou pair, les quantités phi(a) ou chi(a) suivant que a est impair ou pair; on les retranchera si les nombre des facteurs est impair."
%F "To calculate the number of vertices of the polygons with n sides we will get all the divisors a of n that are complementary divisors (codivisors) of the same number having their prime factors at the first power only, and if the number of these factors is even, we will add to phi(n) or chi(n), depending on whether n is even or odd, the quantities phi(a) or chi(a) depending on whether a is odd or even; we will subtract them if the number of factors is odd." - _Michel Marcus_, Feb 03 2013
%t chi[n_] := (8*(2^(n - 1) + 1)*(2^(n - 2) - 1));
%t phi[n_] := (8*(2^(n - 2) + 1)*(2^(n - 1) - 1));
%t either[n_, a_, dsqf_] := (If [Mod[a, 2] == 0, v = chi[a], v = phi[a]]; If [a == n, v, If[Mod[PrimeNu[dsqf], 2] == 0, v, -v]]);
%t picquet[n_] := (ksum = 0; Do[If[SquareFreeQ[d], ksum += either[n, n/d, d]], {d, Divisors[n]}]; ksum/n);
%t Table[picquet[n], {n, 3, 27}] (* _Jean-François Alcover_, Mar 28 2016, after _Michel Marcus_ *)
%o (PARI)
%o chi(n) = {return (8*(2^(n-1)+1)*(2^(n-2)-1));}
%o phi(n) = {return (8*(2^(n-2)+1)*(2^(n-1)-1));}
%o either(n, a, dsqf) = {if ((a % 2) == 0, v = chi(a), v = phi(a)); if (a == n, return (v)); if ((omega(dsqf) % 2) == 0, return (v), return (-v));}
%o picquet(n) = {ksum = 0; fordiv(n, d, if (issquarefree(d), ksum += either(n, n/d, d));); return (ksum/n);}
%o /* _Michel Marcus_, Feb 03 2013 */
%Y Cf. A182589.
%K nonn,nice
%O 3,1
%A _N. J. A. Sloane_
%E Entry revised by _N. J. A. Sloane_, Nov 23 2011
%E More terms from _Michel Marcus_, Feb 03 2013
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