A001616 -id:A001616 - cp's OEIS frontend

A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane

Number of elliptic points of order 3 for Gamma_0(n).
Equivalently, number of fixed points of Gamma_0(n) of type rho.
Values are 0 or a power of 2.
Shadow transform of central polygonal numbers A002061. - Michel Marcus, Jun 06 2013
Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - John M. Campbell, Apr 01 2018
From Jianing Song, Jul 03 2018: (Start)
The comment above is true. Since both a(n) and A001615(n) are multiplicative we just have to verify that for prime powers. Note that A001615(p^e) = (p+1)*p^(e-1). For p == 1 (mod 3), p+1 == 2 (mod 3) so (p+1)*p^(e-1) == 2 (mod 3); for p == 2 (mod 3), p+1 is a multiple of 3 so (p+1)*p^(e-1) == 0 (mod 3). For p = 3, if e = 1 then p+1 == 1 (mod 3); if e > 1 then (p+1)*p^(e-1) == 0 (mod 3).
Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End)

			G.f. = x + x^3 + 2*x^7 + 2*x^13 + 2*x^19 + 2*x^21 + 2*x^31 + 2*x^37 + 2*x^39 + ...

Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).

Christian G. Bower, Table of n, a(n) for n = 1..2000
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. A65 (2009), 156-163. [See Table 4.]
N. J. A. Sloane, Transforms.

Cf. A000089, A000091, A001616, A014683.
Cf. A027748, A079978, A038502, A007949, A165952.
Cf. A341422 (without zeros).

a000086 n = if n `mod` 9 == 0 then 0
  else product $ map ((* 2) . a079978 . (+ 2)) $ a027748_row $ a038502 n
-- Reinhard Zumkeller, Jun 23 2013

with(numtheory); A000086 := proc (n) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end: # Gene Ward Smith, May 22 2006

Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* Michael Somos, Aug 14 2015 *)
a[n_] := a[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, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018 *)

{a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; \\ Nov 15 2002

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 + X, if( p%3==2, 1, (1 + X) / (1 - X)))) [n])}; \\ Nov 15 2002

Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson, Aug 01 2001
a(A226946(n)) = 0; a(A034017(n)) > 0. - Reinhard Zumkeller, Jun 23 2013
a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - Michael Somos, Aug 14 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*sqrt(3)/(3*Pi) = 0.367552... (A165952). - Amiram Eldar, Oct 11 2022

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25

Offset: 1

N. J. A. Sloane, Mar 02 2004

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

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

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.

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
      A127776        4096    1
      A276018        729     1
      A002894        256     1
      A276019        125     4
      A093388        72      1
      A276021        49      9
      A081085        32      1
      A006077        27      1
     A276020        20      2
     A276022        12      1
     A276177        13      36
     A276178        8       1
     A276179        6       1
     A276180        5       4

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 *)

Numbers n such that A001617(n) = 0.

A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2, 1, 3, 3, 3, 1, 2, 4, 3, 3, 3, 5, 3, 4, 3, 5, 4, 3, 1, 2, 5, 5, 4, 4, 5, 5, 5, 6, 5, 7, 4, 7, 5, 3, 5, 9, 5, 7, 7, 9, 6, 5, 5, 8, 5, 8, 7, 11, 6, 7, 4, 9, 7, 11, 7, 10, 9, 9, 7, 11, 7, 10, 9, 11, 9, 9, 7, 7, 9, 7, 8, 15

Offset: 1

N. J. A. Sloane

Also the dimension of the space of cusp forms of weight two and level n. - Gene Ward Smith, May 23 2006

			G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^22 + 2*x^23 + ...

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 1..50000 (first 1000 terms from N. J. A. Sloane)
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Ralf Hemmecke, Peter Paule, and Silviu Radu, Construction of Modular Function Bases for Gamma_0(121) related to p*(11*n + 6), (2019).
Nicolas Allen Smoot, Computer algebra with the fifth operation: applications of modular functions to partition congruences, Ph. D. Thesis, Johannes Kepler Univ., Linz (Austria 2022), 33.
Index entries for sequences related to modular groups

Cf. A001615, A000089, A000086, A001616, A054728, A091401, A091403, A091404.

a := func< n | n lt 1 select 0 else Dimension( CuspForms( Gamma0(n), 2))>; /* Michael Somos, May 08 2015 */

nu2 := proc (n) # number of elliptic points of order two (A000089) local i, s; if modp(n,4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1,i))) fi od; s end:
nu3 := proc (n) # number of elliptic points of order three (A000086) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end:
nupara := proc (n) # number of parabolic cusps (A001616) local b, d; b := 0; for d to n do if modp(n,d) = 0 then b := b+eval(phi(gcd(d,n/d))) fi od; b end:
A001615 := proc(n) local i,j; j := n; for i in divisors(n) do if isprime(i) then j := j*(1+1/i); fi; od; j; end;
genx := proc (n) # genus of X0(n) (A001617) #1+1/12*psi(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: 1+1/12*A001615(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: # Gene Ward Smith, May 23 2006

nu2[n_] := Module[{i, s}, If[Mod[n, 4] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[i] && i > 2, s = s*(1 + JacobiSymbol[-1, i])], {i, Divisors[n]}]; s];
nu3[n_] := Module[{d, s}, If[Mod[n, 9] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[d], s = s*(1 + JacobiSymbol[-3, d])], {d, Divisors[n]}]; s];
nupara[n_] := Module[{b, d}, b = 0; For[d = 1, d <= n, d++, If[ Mod[n, d] == 0, b = b + EulerPhi[ GCD[d, n/d]]]]; b];
A001615[n_] := Module[{i, j}, j = n; Do[ If[ PrimeQ[i], j = j*(1 + 1/i)], {i, Divisors[n]}]; j];
genx[n_] := 1 + (1/12)*A001615[n] - (1/4)*nu2[n] - (1/3)*nu3[n] - (1/2)*nupara[n];
A001617 = Table[ genx[n], {n, 1, 102}] (* Jean-François Alcover, Jan 04 2012, after Gene Ward Smith's Maple program *)
a[ n_] := 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]; (* Michael Somos, May 08 2015 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
vector(102, n, a(n))  \\ Gheorghe Coserea, May 20 2016

a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2.
From Gheorghe Coserea, May 20 2016: (Start)
limsup a(n) / (n*log(log(n))) = exp(Euler)/(2*Pi^2), where Euler is A001620.
a(n) >= (n-5*sqrt(n)-8)/12, with equality iff n = p^2 for prime p=1 (mod 12) (see A068228).
a(n) < n * exp(Euler)/(2*Pi^2) * (log(log(n)) + 2/log(log(n))) for n>=3 (see Csirik link).
(End)

A124315 a(n) = Sum_{ d divides n } tau(gcd(d,n/d)), where tau = sigma_0 = A000005.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 9, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 12, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 18, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 16, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 18, 9, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 24, 2, 8

Offset: 1

Robert G. Wilson v, Sep 30 2006

Apparently the Mobius transform of A046951. - R. J. Mathar, Feb 07 2011

Number of ordered pairs of divisors of n, (d1,d2), with d1<=d2, such that d1|d2 and n|(d1*d2). - Wesley Ivan Hurt, Mar 22 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A000005, A001616, A001620, A046951, A061884, A124316, A306016.

A124315 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do igcd(d,n/d) ; a := a+numtheory[tau](%) ; end do: a; end proc: # R. J. Mathar, Apr 14 2011

Table[Plus @@ Map[DivisorSigma[0, GCD[ #, n/# ]] &, Divisors@n], {n, 98}]
f[p_, e_] := e + 1 + Floor[e^2/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

a(n) = sumdiv(n, d, numdiv(gcd(d, n/d))); \\ Michel Marcus, Feb 12 2016

from sympy import divisors, divisor_count, gcd
def a(n): return sum([divisor_count(gcd(d, n/d)) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

a(p) = 2 iff p is a prime.
Multiplicative with a(p^e) = e+1+floor(e^2/4). - R. J. Mathar, Apr 14 2011
Dirichlet g.f.: zeta(s)^2 * zeta(2*s). - Vaclav Kotesovec, Jan 11 2019
Sum_{k=1..n} a(k) ~ (Pi^2/6) * (n*log(n) + (2*gamma - 1 + 2*zeta'(2)/zeta(2))*n), where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 22 2022

A276183 Genus of the quotient of the modular curve X_0(n) by the Fricke involution.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 3, 0, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 4, 0, 2, 2, 2, 1, 3, 2, 5, 1, 2, 1, 4, 1, 4, 3, 3, 2, 4, 1, 4, 2, 4, 4, 4, 1, 3, 3, 2, 3, 3, 1, 7

Offset: 1

Gheorghe Coserea, Oct 21 2016

nonn

a(n) is the genus of quotient space H/Gamma_0*(n), where H is the upper half plane and Gamma_0*(n) = Gamma_0(n) + W Gamma_0(n) is the extension of Gamma_0(n) via the involution z <-> W(z) = -n/z (see Cohn, 1988).

			G.f. = x^22 + x^28 + x^30 + x^33 + x^34 + x^37 + x^38 + x^40 + 2*x^42 + x^43 + x^44 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..54321
Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.
Harvey Cohn, A Numerical Survey of the Reduction of Modular Curve Genus by Fricke's Involutions, Number Theory (New York Seminar 1989-1990), p. 100.
Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Andrew P. Ogg, Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, vol. 16, no. 1 (1974-1975), talk no. 7, p. 1.

Cf. A000003, A000086, A000089, A001615, A001616, A001617, A276181.

f[n_] := 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];
g[n_] := Ceiling[k0 = k /. FindRoot[EllipticK[1 - k^2]/EllipticK[k^2] == Sqrt@ n, {k, 1/2, 10^-10, 1}, WorkingPrecision -> 600, MaxIterations -> 100]; Exponent[MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2];
r[n_] := If[MemberQ[{3, 7}, #], 3 + (# - 1)/2, 3] &@ Mod[n, 8]; a[n_] := If[n <= 4, 0, (1 + f@ n)/2 - r[n] g[n]/12]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 102}] (* Michael De Vlieger, Oct 28 2016, after Michael Somos at A001617 and Jean-François Alcover at A000003 *)
ClassList[n_?Negative] :=
Select[Flatten[#, 1] &@Table[
    {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}],
  Mod[#3, 1] == 0 && #3 >= # &&
      GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &]
A001617[n_] := 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];
a[n_] := If[0 <= n <= 4, 0, (A001617[n] + 1)/2 - If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]] Length[ClassList[-4 n]]/12] (* David Jao, Sep 07 2020 *)

A000003(n) = qfbclassno(-4*n);
A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k, 1]+1)),
     h = prod(k=1, fsz, f[k, 1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
a(n) = {
  my(r = if (n%8 == 3, 4, n%8 == 7, 6, 3));
  if (n < 5, 0, (1 + A001617(n))/2 - r * A000003(n)/12);
};
vector(102, n, a(n))

a(n) = (1 + A001617(n))/2 - r * A000003(n)/12 for all n > 4, where r=4 for n=3 (mod 8), r=6 for n=7 (mod 8) and r=3 otherwise.

a(n) <> 4884 for all n.

New name from David Jao, Sep 07 2020

A054729 Numbers n such that genus of modular curve X_0(N) is never equal to n.

Original entry on oeis.org

150, 180, 210, 286, 304, 312, 336, 338, 348, 350, 480, 536, 570, 598, 606, 620, 666, 678, 706, 730, 756, 780, 798, 850, 876, 896, 906, 916, 970, 1014, 1026, 1046, 1106, 1144, 1170, 1176, 1186, 1188, 1224, 1244, 1260, 1320, 1350, 1356, 1366

Offset: 1

Janos A. Csirik, Apr 21 2000

nonn

"Looking further in the list of integers not of the form g0(N), we do eventually find some odd values, the first one occurring at the 3885th position. There are four such up to 10^5 (out of 9035 total missed values), namely 49267, 74135, 94091, 96463." (see Csirik link) - Gheorghe Coserea, May 21 2016.

a(1534734) = 9999996. - Gheorghe Coserea, May 23 2016

Gheorghe Coserea, Table of n, a(n) for n = 1..20155
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A054726, A054728, A054730.

a1617[n_] := a1617[n] = 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];
seq[n_] := Module[{inv, bnd}, inv[_] = -1; bnd = 12 n + 18 Floor[Sqrt[n]] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n && inv[g+1] == -1, inv[g+1] = k]]; (Position[Array[inv, n+1], -1] // Flatten)-1];
seq[1000] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
scan(n) = {
  my(inv = vector(n+1,g,-1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n && inv[g+1] == -1, inv[g+1] = k));
  apply(x->(x-1), Vec(select(x->x==-1, inv, 1)))
};
scan(1367)  \\ Gheorghe Coserea, May 21 2016

A061884 a(n) = Sum_{ d | n } phi(lcm(d,n/d)), where phi(n) = Euler totient A000010.

Original entry on oeis.org

1, 2, 4, 5, 8, 8, 12, 12, 14, 16, 20, 20, 24, 24, 32, 26, 32, 28, 36, 40, 48, 40, 44, 48, 44, 48, 48, 60, 56, 64, 60, 56, 80, 64, 96, 70, 72, 72, 96, 96, 80, 96, 84, 100, 112, 88, 92, 104, 90, 88, 128, 120, 104, 96, 160, 144, 144, 112, 116, 160, 120, 120, 168, 116, 192

Offset: 1

Vladeta Jovovic, May 12 2001

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000010, A001616, A057670, A007427.

A061884 := proc(n) local b,d: b := 0; for d from 1 to n do if irem(n,d)=0 then b := b+phi(lcm(d,n/d)); fi; od; RETURN(b); end:

Table[Plus @@ Map[ EulerPhi[LCM[ #, n/# ]] &, Select[ Range@n, (Mod[n, # ] == 0) &]], {n, 65}] (* Robert G. Wilson v, Sep 30 2006 *)

a(n)=sumdiv(n,d,eulerphi(lcm(d,n/d))) \\ Charles R Greathouse IV, Feb 21 2013

A091403 Numbers n such that genus of group Gamma_0(n) is 1.

Original entry on oeis.org

11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49

Offset: 1

N. J. A. Sloane, Mar 02 2004

I assume it is known that there are no further terms? A reference for this would be nice.

Available conductors for modular elliptic curves genus 1. [From Artur Jasinski, Jun 24 2010]

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091404.

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]}];
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]}];
a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
a1616[n_] := Sum[EulerPhi[GCD[ d, n/d]], {d, Divisors[n]}];
a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
Position[Array[a1617, 100], 1] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

Numbers n such that A001617(n) = 1.

A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 12, 12, 12, 16, 14, 16, 24, 16, 18, 24, 20, 24, 32, 24, 24, 32, 30, 28, 36, 32, 30, 48, 32, 32, 48, 36, 48, 48, 38, 40, 56, 48, 42, 64, 44, 48, 72, 48, 48, 64, 56, 60, 72, 56, 54, 72, 72, 64, 80, 60, 60, 96, 62, 64, 96, 64, 84, 96, 68, 72, 96, 96

Offset: 1

Steven Finch, Apr 17 2009

Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
We have
m(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) =
(m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 =
(m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ...
(m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ...
where N is any positive multiple of 4 and j>=1.
Multiplicative because A001615 is multiplicative and a(1) = A001615(2)/3 = 1. - Andrew Howroyd, Aug 08 2018

Ken Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004, (p. 16, theorem 1.56).

Peter Luschny, Table of n, a(n) for n = 1..1000
H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78; Scanned copy.
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014.
Jon Maiga, Computer-generated formulas for A159634, Sequence Machine.
Wikipedia, Cusp Form.

Cf. A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A000010, A000082, A001221, A001615, A008683, A059956, A080512, A159630.

[[4*n,(Dimension(HalfIntegralWeightForms(4*n,7/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2))))/2] : n in [1..70]]; [[4*n,(Dimension(HalfIntegralWeightForms(4*n,5/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,7/2))))/2] : n in [1..70]];

(* per Enrique Pérez Herrero's conjecture proved by P. Humphries, see link *)
dedekindPsi[n_Integer]:=n Apply[Times,1+1/Map[First,FactorInteger[n]]];
1/3 dedekindPsi /@ (2 Range[70]) (* Wouter Meeussen, Apr 06 2014 *)

a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ Andrew Howroyd, Aug 08 2018

a(n) = A159636(n) + A159630(n). - Enrique Pérez Herrero, Apr 15 2014
a(n) = A001615(2*n)/3. - Enrique Pérez Herrero, Jan 31 2014
From Peter Bala, Mar 19 2019: (Start)
a(n)= n*Product_{p|n, p odd prime} (1 + 1/p).
a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function.
If n = m*2^k , where 2^k is the largest power of 2 dividing n, then
a(n) = (2^k)*a(m) = 2^k * Sum_{d^2|m} mu(d)*sigma(m/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
a(n) = 2^k * Sum_{d|m} 2^omega(d)*phi(m/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.
O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End)
a(n) = A000082(n)/A080512(n). [obvious by prime products, discovered by Sequence Machine]. - R. J. Mathar, Jun 24 2021
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = (p+1)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 = 0.607927... (A059956). (End)

A159636 Dimension of space of cusp forms of weight 5/2, level 4*n and trivial character.

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 4, 2, 6, 6, 7, 6, 9, 9, 14, 6, 12, 12, 13, 12, 20, 15, 16, 16, 18, 18, 21, 18, 21, 30, 22, 16, 32, 24, 32, 24, 27, 27, 38, 28, 30, 42, 31, 30, 48, 33, 34, 36, 36, 36, 50, 36, 39, 45, 50, 40, 56, 42, 43, 60

Offset: 1

Steven Finch, Apr 17 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000
H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78.
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]

Cf. A159630, A159634.

[[4*n,Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2)))] : n in [1..75]];

dedekindPsi[n_Integer] := n*Times @@ (1 + 1/First /@ FactorInteger[n]);
\[Chi][n_Integer] := Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
r[(p_)?PrimeQ, n_Integer] := -1+ Last[Flatten[Cases[FactorInteger[p*n], {p, _}]]];
\[Alpha][n_Integer] := Block[{rn}, rn = r[2, n]; If[EvenQ[rn], 3*2^(rn/2 - 1), 2^(rn/2 + 1/2)]];
\[Beta][n_Integer] := Block[{rn}, rn = r[2, n]; Which[rn >= 4, \[Alpha][n], rn === 3, 3, rn === 2 && Or @@ OddQ[(r[#1, n] & ) /@ Select[First /@ FactorInteger[n], Mod[#1, 4] === 3 & ]], 2, True, 3/2]];
s[5/2, n_Integer] := (1/8)* dedekindPsi[n] - \[Beta][n]*(\[Chi][n]/2/\[Alpha][n]);
s[5/2, #] & /@ Range[4, 240, 4] (* Wouter Meeussen, cf. Finch reference, Mar 31 2014 *)

cp's OEIS Frontend

A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

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A091401 Numbers n such that genus of group Gamma_0(n) is zero.

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A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).

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A124315 a(n) = Sum_{ d divides n } tau(gcd(d,n/d)), where tau = sigma_0 = A000005.

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A276183 Genus of the quotient of the modular curve X_0(n) by the Fricke involution.

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A054729 Numbers n such that genus of modular curve X_0(N) is never equal to n.

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A061884 a(n) = Sum_{ d | n } phi(lcm(d,n/d)), where phi(n) = Euler totient A000010.