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%I #10 Oct 19 2018 03:26:01
%S 22,23,26,28,29,31,37,50
%N Numbers n such that genus of group Gamma_0(n) is 2.
%C I assume it is known that there are no further terms? A reference for this would be nice.
%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
%D G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.
%F Numbers n such that A001617(n) = 2.
%t a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
%t a86[n_] := a86[n] = Product[{p, e} = pe; Which[p == 1 || p == 3 && e == 1, 1, p == 3 && e > 1, 0, Mod[p, 3] == 1, 2, Mod[p, 3] == 2, 0, True, a86[p^e]], {pe, FactorInteger[n]}];
%t a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
%t a1616[n_] := Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
%t a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
%t Position[Array[a1617, 100], 2] // Flatten (* _Jean-François Alcover_, Oct 19 2018 *)
%Y Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091403.
%K nonn,fini,full
%O 1,1
%A _N. J. A. Sloane_, Mar 02 2004
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