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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.

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

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Last modified July 29 06:36 EDT 2021. Contains 346340 sequences. (Running on oeis4.)