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

1,2

Comments

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

References

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

Links

Table of n, a(n) for n=1..15.

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.

Formula

Numbers n such that A001617(n) = 0.

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 *)

Crossrefs

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

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

Sequence in context: A011875 A249575 A053433 * A278581 A191889 A091402

Adjacent sequences: A091398 A091399 A091400 * A091402 A091403 A091404

Keyword

nonn,fini,full

Author

N. J. A. Sloane, Mar 02 2004

Status

approved