A000089 -id:A000089 - cp's OEIS frontend

A273445 a(n) is the number of solutions of the equation n = A001617(k).

15, 12, 8, 11, 7, 14, 4, 13, 7, 12, 4, 15, 4, 9, 6, 10, 5, 16, 2, 20, 3, 14, 7, 11, 2, 13, 5, 11, 3, 14, 3, 9, 6, 13, 3, 17, 3, 14, 4, 10, 4, 20, 3, 15, 3, 12, 1, 15, 2, 20, 4, 11, 3, 13, 3, 16, 3, 12, 3, 15, 3, 12, 5, 9, 4, 15, 2, 14, 5, 17, 3, 13

Offset: 0

Gheorghe Coserea, May 22 2016

nonn

The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.

			For n = 0 the a(0) = 15 solutions are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).
For n = 1 the a(1) = 12 solutions are:
11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).
For n = 2 the a(2) = 8 solutions are:
22, 23, 26, 28, 29, 31, 37, 50 (A091404).

Gheorghe Coserea, Table of n, a(n) for n = 0..100001
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054729, A091401, A091403, A091404.

(* b = A001617 *) nmax = 71;
b[n_] := b[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];
Clear[f];
f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];
f[m = nmax]; f[m = m + nmax];
While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];
A273445 = f[m] (* Jean-François Alcover, Dec 16 2018, using Michael Somos' code for 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;
seq(n) = {
  my(a = vector(n+1,g,0), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n, a[g+1]++));
  return(a);
};
seq(72)

a(n) = card {k, n = A001617(k)}.

A350871 Number of well-rounded sublattices of index n in square lattice.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 4, 3, 2, 0, 0, 2, 2, 0, 1, 0, 2, 2, 1, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 8, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 0, 1, 2, 2, 4, 0, 2, 0, 0, 6, 1, 2, 0, 2, 4, 0, 0

Offset: 1

Andrey Zabolotskiy, Jan 20 2022

nonn

A sublattice is well-rounded if the linear span of its vectors of minimal length is the whole space.
A sublattice of the square lattice is well-rounded when it is square or centered rectangular (rhombic) with not too oblong unit cell: the angles of the rhombus should be at least Pi/3.
In this sequence, any two sublattices differing by any isometry are counted as distinct.

			a(4) = 1 well-rounded index-4 sublattice has basis (2, 0), (0, 2).
a(5) = 2 w.-r. index-5 sublattices have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).
At index 12, for the first time centered rectangular sublattices occur, there are a(12) = 2 of them with the bases (3, 2), (3, -2) and (2, 3), (-2, 3).

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Michael Baake and Peter Zeiner, Geometric enumeration problems for lattices and embedded Z-modules, arXiv:1709.07317 [math.MG], 2017; in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172. See table "Some counts of the enumeration problems for Z^2"; beware of the typo in the 60th term.
Peter Zeiner, Coincidence Site Lattices and Coincidence Site Modules, Thesis, Universität Bielefeld, 2015.
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. enumeration of other classes of sublattices of Z^2: A000203 (all sublattices), A002654 (square sublattices), A000089 (primitive square sublattices), A350872 (coincidence sublattices), A145393 (all sublattices up to isometries of the parent lattice).

Cf. A097584.

fa[s_] := Count[Divisors[s], _?(#^2 < (s/#)^2 < 3 #^2 &)];
f0[s_] := If[OddQ[s], 0, 2 fa[s/2]];
f1[s_] := With[{e2 = IntegerExponent[s, 2]}, 2 (-1)^e2 fa[s/2^e2]];
pr[s_] := Count[Range[s], _?(Mod[#^2 + 1, s] == 0 &)]; (*A000089*)
sq[n_] := Sum[pr[n/d^2], {d, Select[Range[n], Mod[n, #^2] == 0 &]}]; (*A002654*)
a[n_] := sq[n] + Sum[pr[n/d] (f0[d] + f1[d]), {d, Divisors[n]}];
Array[a, 87]

See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]

A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 6, 1, 1, 3, 2, 3, 1, 3, 2, 1, 3, 1, 2, 1, 1, 3, 6, 1, 1, 3, 6, 1, 1, 1, 6, 3, 1, 3, 2, 3, 3, 3, 2, 1, 1, 3, 2, 1, 1, 1, 6, 3, 3, 3, 2, 3, 1, 1, 6, 1, 3, 3, 2, 1, 1, 9, 2, 1, 3, 1, 6, 1, 1, 3, 6, 3, 1, 1, 6, 1, 3, 1, 6, 1, 1, 9, 2, 3, 1, 3, 6, 3, 1, 1, 2, 3, 1, 3, 2, 1, 3, 3, 6, 1, 3, 3

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Shadow transform of A034262. - Michel Marcus, Jun 06 2013

Andrew Howroyd, Table of n, a(n) for n = 1..10000
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.
N. J. A. Sloane, Transforms.

Cf. A087688, A000089, A034262.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[p == 2, If[e == 1, 2, 1], If[Mod[p, 4] == 1, 3, 1]], {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019 *)

a(n)={my(v=vector(n)); sum(i=0, n-1, lift(Mod(i,n)^3 + i) == 0)} \\ Andrew Howroyd, Jul 15 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, if(e==1, 2, 1), if(p%4==1, 3, 1)))} \\ Andrew Howroyd, Jul 15 2018

Multiplicative with a(2^1) = 2, a(2^e) = 1 for e > 1, a(p^e) = 3 for p mod 4 == 1, a(p^e) = 1 for p mod 4 == 3. - Andrew Howroyd, Jul 15 2018

More terms from David Wasserman, Jun 17 2005

A185278 Number of isomorphism classes of generalized Petersen graphs G(n,k) on 2n vertices with gcd(n,k) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 2, 3, 3, 5, 2, 5, 3, 4, 3, 6, 4, 6, 4, 5, 4, 8, 3, 8, 5, 6, 5, 7, 4, 10, 5, 7, 6, 11, 4, 11, 6, 7, 6, 12, 6, 11, 6, 9, 7, 14, 5, 11, 8, 10, 8, 15, 6, 16, 8, 10, 9, 14, 6, 17, 9, 12, 7, 18, 8, 19, 10, 11, 10, 16, 7, 20, 10, 14, 11, 21, 8, 18, 11, 15, 12, 23, 7, 19, 12, 16, 12, 19, 10, 25, 11, 16, 11

Offset: 3

N. J. A. Sloane, Feb 19 2011

Robert Israel, Table of n, a(n) for n = 3..10000
Marko Petkovsek and Helena Zakrajsek, Enumeration of I-graphs: Burnside does it again, Ars Mathematica Contemporanea, 2 (2009) 241-262.
A. Steimle and W. Staton, The isomorphism classes of the generalized Petersen graphs, Discrete Math. 309 (2009), 231-237.

Cf. A000010, A000089, A060594.

# using functions A060594 and A000089 as defined in those sequences
f:= n -> (numtheory:-phi(n)+A060594(n)+A000089(n))/4:
map(f, [$3..100]); # Robert Israel, Sep 06 2018

a(n) = (A000010(n) + A060594(n) + A000089(n))/4.

A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

Original entry on oeis.org

25, 49, 50, 64, 81, 75, 121, 100, 169, 128, 127, 147, 157, 163, 181, 193, 199, 289, 229, 243, 239, 257, 361, 283, 293, 313, 343, 337, 349, 353, 373, 379, 397, 409, 421, 529, 439, 457, 463, 467, 487, 499, 509, 523, 541, 547, 557, 577, 625, 601, 613, 619, 631, 643, 661, 673, 677, 691, 841, 667, 733

Offset: 0

Gheorghe Coserea, May 23 2016

sign

a(10^7) = 120000007 is the largest value in the first 1+10^7 terms of the sequence.

The exception occurs first at a(150) = -1. - Georg Fischer, Feb 15 2019

			For n = 0 we have 0 = A001617(k) when k is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401); the largest of this values is 25 therefore a(0) = 25.
For n = 1 we have 1 = A001617(k) when k is 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403); the largest of this values is 49 therefore a(1) = 49.
For n = 2 we have 2 = A001617(k) when k is 22, 23, 26, 28, 29, 31, 37, 50 (A091404); the largest of this values is 50 therefore a(2) = 50.
For n = 150 (= A054729(1)) we have 150 <> A001617(k) for all k therefore a(150) = -1.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054728, A054729, A091401, A091403, A091404, A273445.

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[{a, bnd}, a = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n] ] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n, a[[g+1]] = k]]; a];
seq[60] (* 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;
seq(n) = {
  my(a = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k); if (g <= n, a[g+1] = k));
  return(a);
};
seq(60)

Let S(n) = {k, n = A001617(k)}; if the level set S(n) is not empty then a(n) = max S(n) and A054728(n) = min S(n) and A273445(n) = card S(n), otherwise a(n) = A054728(n) = -1 and A273445(n) = 0.

A276181 Fricke's 37 cases for two-valued modular equations.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59, 71

Offset: 1

Gheorghe Coserea, Oct 17 2016

Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.

Cf. A000003, A001617.

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;
A276183(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);
};
select(x->(x>1), Vec(select(x->x==0, vector(100, n, A276183(n)), 1)))

Numbers n>1 such that 0 = A276183(n).

A385104 Triangle read by rows: T(n,k) is the number of residue classes obtained by solving mod(x^2,n) = k for x over the integers, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 2, 0, 1, 2, 0, 1, 2, 2, 0, 2, 0, 0, 2, 4, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 1, 2, 0, 0, 2, 1, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 2, 0, 2, 0, 0, 1, 2

Offset: 1

Jason Bard, Jun 18 2025

The sum of each row is n.

			Triangle starts:
  1
  1 1
  1 2 0
  2 2 0 0
  1 2 0 0 2
  1 2 0 1 2 0
  1 2 2 0 2 0 0
  2 4 0 0 2 0 0 0
  3 2 0 0 2 0 0 2 0
  1 2 0 0 2 1 2 0 0 2
  ...

Jason Bard, Table of n, a(n) for n = 1..11325

Cf. A000089, A000188, A060594, A096008.

dat[n_] := Table[Reduce[Mod[x^2, n] == k, x, Integers], {k, 0, n - 1}]; countConditions[cond_] := Which[cond === False, 0, MatchQ[cond, x \[Element] Integers], 1, True, Length@Cases[cond, Equal[x, _], Infinity]]; counts = Flatten[Table[countConditions /@ dat[n], {n, 1, 20}]]

T(n, k) = sum(i=1, n, Mod(i,n)^2 == k);
row(n) = vector(n, i, T(n, i-1)); \\ Michel Marcus, Jun 23 2025

T(n,0) = A000188(n).
T(n,1) = A060594(n).
T(n,n-1) = A000089(n).

A087785 Number of elements in GL(2,Z_n) x with x^2 == -I mod n where I is the identity matrix.

Original entry on oeis.org

1, 4, 6, 12, 32, 24, 42, 48, 54, 128, 110, 72, 184, 168, 192, 192, 308, 216, 342, 384, 252, 440, 506, 288, 752, 736, 486, 504, 872, 768, 930, 768, 660, 1232, 1344, 648, 1408, 1368, 1104, 1536, 1724, 1008, 1806, 1320, 1728, 2024, 2162, 1152, 2058, 3008, 1848

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Cf. A066907, A000089.

More terms from David Wasserman, Jun 17 2005

A276184 Numbers n such that A276183(n) = 1.

Original entry on oeis.org

22, 28, 30, 33, 34, 37, 38, 40, 43, 44, 45, 48, 51, 53, 54, 55, 56, 61, 63, 64, 65, 75, 79, 81, 83, 89, 95, 101, 119, 131

Offset: 1

Gheorghe Coserea, Oct 22 2016

Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.

Cf. A276183.

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;
A276183(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);
};
Vec(select(x->x==1, vector(5000, n, A276183(n)), 1))

A276185 Numbers n such that A276183(n) = 2.

Original entry on oeis.org

42, 46, 52, 57, 62, 67, 68, 69, 72, 73, 74, 77, 80, 87, 91, 98, 103, 107, 111, 121, 125, 143, 167, 191

Offset: 1

Gheorghe Coserea, Oct 22 2016

Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.

Cf. A276183.

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;
A276183(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);
};
Vec(select(x->x==2, vector(500, n, A276183(n)), 1))

cp's OEIS Frontend

A273445 a(n) is the number of solutions of the equation n = A001617(k).

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A350871 Number of well-rounded sublattices of index n in square lattice.

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A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).

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Extensions

A185278 Number of isomorphism classes of generalized Petersen graphs G(n,k) on 2n vertices with gcd(n,k) = 1.

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A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

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A276181 Fricke's 37 cases for two-valued modular equations.

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A385104 Triangle read by rows: T(n,k) is the number of residue classes obtained by solving mod(x^2,n) = k for x over the integers, n >= 1, k >= 0.

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Formula

A087785 Number of elements in GL(2,Z_n) x with x^2 == -I mod n where I is the identity matrix.