A000086 -id:A000086 - cp's OEIS frontend

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.

A376202 Number of pairs 1 <= x <= y <= n-1 such that gcd(x,n) = gcd(y,n) = gcd(x+y,n) = 1 and 1/x + 1/y == 1/(x+y) mod n.

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 36, 0, 24, 0, 0, 0, 42, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 60, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 60, 0, 0, 0, 96, 0, 0, 0, 0, 0, 102, 0, 0

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

In general, 1/x + 1/y = 1/(x+y) is the wrong way to add fractions!
See A376203 for a(2*n-1)/2 and A376755 for a(6*n+1)/6.
From Robert Israel, Nov 06 2024: (Start)
If a(n) = 0 then a(m) = 0 whenever m is a multiple of n.
It appears that the primes p for which a(p) > 0 are A007645. (End)

			For n = 3 the a(3) = 2 solutions are (x,y) = (1,1) and (2,2).
For n = 7 the a(7) = 6 solutions are (x,y) = (1,2), (1,4), (2,4), (3,5), (3,6), (5,6).

Robert Israel, Table of n, a(n) for n = 1..6000

Cf. A000086, A007645, A046530, A290731, A376203, A376755, A376756, A376757.

a:=[];
for n from 1 to 140 do
c:=0;
for y from 1 to n-1 do
for x from 1 to y do
if gcd(y,n) = 1 and gcd(x,n) = 1 and gcd(x+y,n) = 1  and (1/x + 1/y - 1/(x+y)) mod n = 0 then c:=c+1; fi;
od: # od x
od: # od y
a:=[op(a),c];
od: # od n
a;

from math import gcd
def A376202(n):
    c = 0
    for x in range(1,n):
        if gcd(x,n) == 1:
            for y in range(x,n):
                if gcd(y,n)==gcd(z:=x+y,n)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%n:
                    c += 1
    return c # Chai Wah Wu, Oct 06 2024

A376203 a(n) = A376202(2*n-1)/2.

Original entry on oeis.org

0, 1, 0, 3, 0, 0, 6, 0, 0, 9, 6, 0, 0, 0, 0, 15, 0, 0, 18, 12, 0, 21, 0, 0, 21, 0, 0, 0, 18, 0, 30, 0, 0, 33, 0, 0, 36, 0, 0, 39, 0, 0, 0, 0, 0, 72, 30, 0, 48, 0, 0, 51, 0, 0, 54, 36, 0, 0, 0, 0, 0, 0, 0, 63, 42, 0, 108, 0, 0, 69

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..7814

Cf. A000086, A087786, A046530, A290731, A376202, A376755, A376756, A376757.

from math import gcd
def A376203(n):
    c, m = 0, (n<<1)-1
    for x in range(1,m):
        if gcd(x,m) == 1:
            for y in range(x,m):
                if gcd(y,m)==gcd(z:=x+y,m)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%m:
                    c += 1
    return c>>1 # Chai Wah Wu, Oct 06 2024

A376755 a(n) = A376202(6*n+1)/6.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, 7, 0, 10, 11, 12, 13, 0, 24, 16, 17, 18, 0, 0, 21, 36, 23, 0, 25, 26, 27, 26, 0, 30, 0, 32, 33, 0, 35, 60, 37, 38, 0, 40, 72, 0, 72, 0, 45, 46, 47, 0, 0, 84, 51, 52, 0, 0, 55, 56, 49, 58, 0, 57, 61, 62, 63, 0, 0, 66, 120, 68, 0, 70, 120, 72

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..3470

Cf. A000086, A046530, A087786, A290731, A376202, A376203, A376756, A376757.

from math import gcd
def A376755(n):
    c, m = 0, 6*n|1
    for x in range(1,m):
        if gcd(x,m) == 1:
            for y in range(x,m):
                if gcd(y,m)==gcd(z:=x+y,m)==1 and not (w:=z**2-x*y)//gcd(w,x*y*z)%m:
                    c += 1
    return c//6 # Chai Wah Wu, Oct 06 2024

a(51)-a(72) from Chai Wah Wu, Oct 06 2024

A376756 Number of pairs 0 <= x <= y <= n-1 such that x^2 + x*y + y^2 == 0 (mod n).

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 7, 3, 6, 1, 1, 9, 13, 7, 3, 10, 1, 6, 19, 3, 21, 1, 1, 9, 15, 13, 18, 27, 1, 3, 31, 10, 3, 1, 7, 21, 37, 19, 39, 3, 1, 21, 43, 3, 6, 1, 1, 30, 70, 15, 3, 51, 1, 18, 1, 27, 57, 1, 1, 9, 61, 31, 60, 36, 13, 3, 67, 3, 3, 7, 1, 21, 73, 37, 45, 75, 7, 39, 79, 10, 45, 1, 1, 81, 1, 43, 3, 3, 1, 6, 163, 3, 93, 1, 19, 30, 97

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000086, A046530, A087786, A290731, A376202, A376203, A376755, A376757.

def A376756(n):
    c = 0
    for x in range(n):
        z = x**2%n
        for y in range(x,n):
            if not (z+y*(x+y))%n:
                c += 1
    return c # Chai Wah Wu, Oct 06 2024

A376757 Number of pairs 0 <= x <= y <= n-1 such that x^3 == y^3 (mod n).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 13, 14, 18, 10, 11, 15, 25, 26, 15, 28, 17, 36, 37, 25, 39, 22, 23, 42, 35, 50, 81, 71, 29, 30, 61, 72, 33, 34, 65, 99, 73, 74, 75, 70, 41, 78, 85, 55, 90, 46, 47, 84, 112, 70, 51, 137, 53, 162, 55, 218, 111, 58, 59, 75, 121, 122, 288, 208, 125, 66, 133, 85, 69, 130, 71, 306, 145, 146, 105, 203, 143, 150, 157

Offset: 1

Tom Duff and N. J. A. Sloane, Oct 06 2024

nonn

A087786 includes pairs (x,y) with x>y (which are excluded from the present sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000086, A087786, A046530, A290731, A376202, A376203, A376755, A376756.

a(n) = sum(x=0, n-1, sum(y=x, n-1, Mod(x, n)^3 == Mod(y, n)^3)); \\ Michel Marcus, Oct 06 2024

from collections import Counter
def A376757(n): return sum(d*(d+1)>>1 for d in Counter(pow(x,3,n) for x in range(n)).values()) # Chai Wah Wu, Oct 06 2024

A054728 a(n) is the smallest 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

1, 11, 22, 30, 38, 42, 58, 60, 74, 66, 86, 78, 106, 105, 118, 102, 134, 114, 223, 132, 166, 138, 188, 156, 202, 168, 214, 174, 236, 186, 359, 204, 262, 230, 278, 222, 298, 240, 314, 246, 326, 210, 346, 270, 358, 282, 557, 306, 394, 312, 412, 318

Offset: 0

Janos A. Csirik, Apr 21 2000

sign

a(150) = -1, a(n) > 0 for 0<=n<=149.

a(9999988) = 119999861 is the largest value in the first 1+10^7 terms of the sequence. - Gheorghe Coserea, May 24 2016

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054727, A054729.

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 = Table[-1, {n+1}]; bnd = 12n + 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]]; inv];
seq[51] (* 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(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));
  return(inv);
};
seq(51)  \\ Gheorghe Coserea, May 21 2016

A001617(a(A001617(n))) = A001617(n) and a(A054729(n)) = -1 for all n>=1. - Gheorghe Coserea, May 22 2016

A091404 Numbers n such that genus of group Gamma_0(n) is 2.

Original entry on oeis.org

22, 23, 26, 28, 29, 31, 37, 50

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.

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, A091403.

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], 2] // Flatten (* Jean-François Alcover, Oct 19 2018 *)

Numbers n such that A001617(n) = 2.

A000091 Multiplicative with a(2^e) = 2 for k >= 1; a(3) = 2, a(3^e) = 0 for k >= 2; a(p^e) = 0 if p > 3 and p == -1 (mod 3); a(p^e) = 2 if p > 3 and p == 1 (mod 3).

Original entry on oeis.org

1, 2, 2, 2, 0, 4, 2, 2, 0, 0, 0, 4, 2, 4, 0, 2, 0, 0, 2, 0, 4, 0, 0, 4, 0, 4, 0, 4, 0, 0, 2, 2, 0, 0, 0, 0, 2, 4, 4, 0, 0, 8, 2, 0, 0, 0, 0, 4, 2, 0, 0, 4, 0, 0, 0, 4, 4, 0, 0, 0, 2, 4, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 4, 0, 4, 0, 8, 2, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, 4, 2, 4, 0, 0, 0, 0, 2, 4, 0

Offset: 1

N. J. A. Sloane

Christian G. Bower, Table of n, a(n) for n = 1..2000

A000091 := proc(n) local b,d,nt,c; if n = 1 then RETURN(1); fi; c := 1; if n mod 2 = 0 then c := c*2; fi; if n mod 3 = 0 then c := c*2; fi; nt := n; while nt mod 2 = 0 do nt := nt/2; od; while nt mod 3 = 0 do nt := nt/3; od; if irem(n,9) = 0 then RETURN(0); fi; b := 1; for d from 3 to nt do if irem(nt,d) = 0 and isprime(d) then b := b*(1+legendre(-3,d)); fi; od; RETURN(b*c); end;

a[1] = 1; a[n_] := Block[{b, d, nt, c = 1}, If[Mod[n, 2] == 0, c = c*2]; If[Mod[n, 3] == 0, c = c*2]; nt = n; While[ Mod[nt, 2] == 0, nt = nt/2]; While[ Mod[nt, 3] == 0, nt = nt/3]; If[Mod[n, 9] == 0, Return[0]]; b = 1; For[d = 3, d <= nt, d++, If[Mod[nt, d] == 0 && PrimeQ[d], b = b*(1+JacobiSymbol[-3, d])]]; Return[b*c]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 06 2012, after Maple *)

Description corrected Mar 02 2004. (The old description defined A000086, not this sequence.)

A226946 Numbers that can't be written as x^2 + x*y + y^2, with 0 <= x <= y and gcd(x,y) = 1.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Reinhard Zumkeller, Jun 23 2013

nonn

A000086(a(n)) = 0.

Also numbers n with psi(n) congruent to 0 mod 3, where psi is A001615, and also numbers divisible by 9 or by at least one prime of the form 3k-1: A003627. - Enrique Pérez Herrero, Dec 08 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A034017 (complement).

Cf. A003627, A001615.

a226946 n = a226946_list !! (n-1)
a226946_list = filter ((== 0) . a000086) [1..]
-- Reinhard Zumkeller, Dec 16 2013, Jun 23 2013

Select[Range[1000],Mod[# * Times@@(1+1/Transpose[FactorInteger[#]][[1]]),3]==0&] (* Enrique Pérez Herrero, Dec 08 2013 *)
nn = 10; lim = 1 + nn + nn^2; Complement[Range[2, lim], Select[Union[Flatten[Table[If[GCD[x, y] == 1, x^2 + x*y + y^2, 0], {y, nn}, {x, y}]]], # <= lim &]] (* T. D. Noe, Dec 09 2013 *)

The original definition was too weak; thanks to T. D. Noe for having corrected this. - Reinhard Zumkeller, Dec 09 2013

cp's OEIS Frontend

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

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A376202 Number of pairs 1 <= x <= y <= n-1 such that gcd(x,n) = gcd(y,n) = gcd(x+y,n) = 1 and 1/x + 1/y == 1/(x+y) mod n.

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A376203 a(n) = A376202(2*n-1)/2.

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A376755 a(n) = A376202(6*n+1)/6.

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A376756 Number of pairs 0 <= x <= y <= n-1 such that x^2 + x*y + y^2 == 0 (mod n).

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A376757 Number of pairs 0 <= x <= y <= n-1 such that x^3 == y^3 (mod n).

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PARI

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A054728 a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

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A091404 Numbers n such that genus of group Gamma_0(n) is 2.

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A000091 Multiplicative with a(2^e) = 2 for k >= 1; a(3) = 2, a(3^e) = 0 for k >= 2; a(p^e) = 0 if p > 3 and p == -1 (mod 3); a(p^e) = 2 if p > 3 and p == 1 (mod 3).