A091401 Numbers n such that genus of group Gamma_0(n) is zero.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25
Offset: 1
References
- G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.
Links
- Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3; See Eq. (22). - _N. J. A. Sloane_, Jun 19 2014
- K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 15.
- Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
- Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
- K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.
- B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
Crossrefs
The table below is a consequence of Theorem 7.3 in Maier's paper.
N EntryID K alpha
1
2 A127776 4096 1
3 A276018 729 1
4 A002894 256 1
5 A276019 125 4
6 A093388 72 1
7 A276021 49 9
8 A081085 32 1
9 A006077 27 1
10 A276020 20 2
12 A276022 12 1
13 A276177 13 36
16 A276178 8 1
18 A276179 6 1
25 A276180 5 4
Programs
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Mathematica
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 *)
Formula
Numbers n such that A001617(n) = 0.
Comments