

A091401


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


22



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
YangHui 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 qSeries, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.
B. Schoeneberg, Elliptic Modular Functions, SpringerVerlag, 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



