login
Numbers n such that genus of group Gamma_0(n) is zero.
22

%I #32 Dec 05 2016 11:29:38

%S 1,2,3,4,5,6,7,8,9,10,12,13,16,18,25

%N Numbers n such that genus of group Gamma_0(n) is zero.

%C Equivalently, numbers n such that genus of modular curve X_0(n) is zero.

%D G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

%H Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, <a href="http://dx.doi.org/10.1186/2197-9847-1-3">Umbral moonshine and the Niemeier lattices</a>, Research in the Mathematical Sciences, 2014, 1:3; See Eq. (22). - _N. J. A. Sloane_, Jun 19 2014

%H K. Harada, <a href="http://dx.doi.org/10.4171/090">"Moonshine" of Finite Groups</a>, European Math. Soc., 2010, p. 15.

%H Yang-Hui He, John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.

%H Robert S. Maier, <a href="http://arxiv.org/abs/math/0611041">On Rationally Parametrized Modular Equations</a>, arXiv:math/0611041 [math.NT], 2006.

%H K. Ono, <a href="http://bookstore.ams.org/cbms-102">The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series</a>, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.

%H B. Schoeneberg, <a href="http://dx.doi.org/10.1007/978-3-642-65663-7">Elliptic Modular Functions</a>, Springer-Verlag, NY, 1974, p. 103.

%F Numbers n such that A001617(n) = 0.

%t Flatten@ Position[#, 0] &@ Table[If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@ n}] - Count[(#^2 - # + 1)/n & /@ Range@ n, _?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range@ n, _?IntegerQ]/4], {n, 120}] (* _Michael De Vlieger_, Dec 05 2016, after _Michael Somos_ at A001617 *)

%Y Cf. A001617, A001615, A000089, A000086, A001616, A091403.

%Y The table below is a consequence of Theorem 7.3 in Maier's paper.

%Y N EntryID K alpha

%Y 1

%Y 2 A127776 4096 1

%Y 3 A276018 729 1

%Y 4 A002894 256 1

%Y 5 A276019 125 4

%Y 6 A093388 72 1

%Y 7 A276021 49 9

%Y 8 A081085 32 1

%Y 9 A006077 27 1

%Y 10 A276020 20 2

%Y 12 A276022 12 1

%Y 13 A276177 13 36

%Y 16 A276178 8 1

%Y 18 A276179 6 1

%Y 25 A276180 5 4

%K nonn,fini,full

%O 1,2

%A _N. J. A. Sloane_, Mar 02 2004