A079896 -id:A079896 - cp's OEIS frontend

A361166 Number of elliptic points of order 2 in Weierstrass curve with discriminant A079896(n) in moduli space M_4 of compact Riemann surfaces of genus 4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 14 2023

McMullen, Curtis. "Billiards and Teichmüller curves." Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.3.

Cf. A079896, A361165, A361167, A361168.

A361168 Number of cusps in Weierstrass curve with discriminant A079896(n) in moduli space M_4 of compact Riemann surfaces of genus 4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 14 2023

McMullen, Curtis. "Billiards and Teichmüller curves." Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.3.

Cf. A079896, A361165, A361166, A361167.

A226693 Squarefree parts of A079896(n), n>= 1.

Original entry on oeis.org

5, 2, 3, 13, 17, 5, 21, 6, 7, 29, 2, 33, 37, 10, 41, 11, 5, 3, 13, 53, 14, 57, 15, 61, 65, 17, 69, 2, 73, 19, 77, 5, 21, 85, 22, 89, 23, 93, 6, 97, 101, 26, 105, 3, 109, 7, 113, 29, 13, 30, 31, 5, 2, 129, 33, 133, 34, 137, 35, 141, 145, 37, 149, 38, 17, 39, 157, 10

Offset: 1

Wolfdieter Lang, Jun 15 2013

a(n) is the squarefree part of the discriminant D(n) = A079896(n) of indefinite binary quadratic forms. Certain quadratic irrationals, called omega_p(D(n)), related to the principal indefinite form of discriminant D(n) are integers in the quadratic number field Q(sqrt(a(n))). See A226166 for the definition of these irrationals omega_p(D(n)) using the D. A. Buell reference, p. 31 and p. 26.

For discriminants D == 1 (mod 4) these squarefree parts are given in A226165. For D == 0 (mod 4) the squarefree parts are given in A002734 corresponding to A000037 = D/4.

D. A. Buell, Binary Quadratic Forms, Springer, 1989.

Gheorghe Coserea, Table of n, a(n) for n = 1..20001

Cf. A079896, A226165, A002734.

SquareFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]); SquareFreePart /@ Select[ Range[160], ! IntegerQ[Sqrt[#]] && Mod[#, 4] < 2 &] (* Jean-François Alcover, Jun 25 2013 *)

A079896_list(N) = {
  my(n = 1, v = vector(N), top = 0);
  while (top < N, if (n%4 < 2 && !issquare(n), v[top++] = n); n++;);
  return(v);
};
apply(core, A079896_list(68)) \\ Gheorghe Coserea, Nov 10 2016

a(n) = squarefree part of D(n) = A079896(n), n >= 1, the numbers 0 and 1 (mod 4), not a square.

Offset corrected by Robin Visser, Jun 01 2025

A267857 Length of the period of the continued fraction for the square root of D, the discriminant of indefinite binary quadratic forms. D is given in A079896.

Original entry on oeis.org

1, 2, 2, 5, 1, 2, 6, 2, 4, 5, 4, 4, 1, 2, 3, 8, 6, 2, 6, 5, 2, 6, 4, 11, 1, 2, 8, 2, 7, 12, 6, 2, 2, 5, 6, 5, 8, 10, 4, 11, 1, 2, 2, 8, 15, 6, 9, 10, 6, 2, 16, 5, 4, 10, 2, 16, 4, 9, 4, 4, 1, 2, 9, 2, 8, 2, 17, 8, 10, 6, 6, 2, 16, 5, 4, 8, 4, 21

Offset: 1

Wolfdieter Lang, Feb 03 2016

This is a subsequence of A003285.
If a(n) is even then the smallest positive integer solution of the Pell equation x^2 - D(n)*y^2 = +1 with D(n) = A079896(n) is given by (x0, y0) = (P,Q) with P/Q = [a,b[1], ..., b[a(n)-1]]. If a(n) is odd then the smallest positive integer solution of the Pell equation x^2 - D(n)*y^2 = +1 is given by (x0, y0) = (P^2 + D(n)*Q^2, 2*P*Q). See e.g., the Silverman reference Theorem 40.4 on p. 351.
For positive integer d, d not a square, the Pell equations X^2 - d*Y^2 = +4 and X^2 - d*Y^2 = -4 have no proper solutions. For D(n) = A079896(n) there are solutions for X^2 - D(n)*Y^2 = +4 or -4 (inclusive or). See the Wolfdieter Lang link under A225953 for Pell +4 or -4 solutions.

			a(1)  = 1  because sqrt(5)  = [2,repeat(4)].
a(2)  = 2  because sqrt(8)  = [2,repeat(1,4)].
a(24) = 11 because sqrt(61) = [7,repeat(1,4,3,1,2,2,1,3,4,1,14)].
Pell +1 equation: n = 24 with D = 61 has odd a(24)
  P/Q = [7,1,4,3,1,2,2,1,3,4,1] = 29718/3805 (in lowest terms). Therefore (x0, y0) = (1766319049, 226153980), see A174762 (Of course, (1, 0) is the smallest nonnegative solution.)

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 351.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A003285, A033313, A033317, A079896, A225953.

See the Robert Israel program under A003285, adapted to n -> a(n).

Length[Last@ #] & /@ ContinuedFraction@ Sqrt@ Select[Range@ 200, And[MemberQ[{0, 1}, Mod[#, 4]], ! IntegerQ@ Sqrt@ #] &] (* Michael De Vlieger, Feb 11 2016, after A079896 *)

def a(n):
    i, D = 1, Integer(5)
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    K. = QuadraticField(D)
    return continued_fraction(a).period_length()  # Robin Visser, Jun 06 2025

Offset corrected by Robin Visser, Jun 06 2025

A361157 Genus of Weierstrass curve with discriminant A079896(n) in moduli space M_2 of compact Riemann surfaces of genus 2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 14 2023

Curtis McMullen, Billiards and Teichmüller curves, Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.1.

Cf. A079896, A361158, A361159.

A361158 Number of elliptic points of order 2 in Weierstrass curve with discriminant A079896(n) in moduli space M_2 of compact Riemann surfaces of genus 2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 14 2023

McMullen, Curtis. "Billiards and Teichmüller curves." Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.1.

Cf. A079896, A361157, A361159.

A361159 Number of cusps in Weierstrass curve with discriminant A079896(n) in moduli space M_2 of compact Riemann surfaces of genus 2.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 4, 6, 7, 5, 7, 6, 9, 12, 7, 9, 8, 11, 15, 7, 10, 10, 12

Offset: 1

N. J. A. Sloane, Mar 14 2023

McMullen, Curtis. "Billiards and Teichmüller curves." Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.1.

Cf. A079896, A361157, A361158.

A361167 Number of elliptic points of order 3 in Weierstrass curve with discriminant A079896(n) in moduli space M_4 of compact Riemann surfaces of genus 4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 14 2023

McMullen, Curtis. "Billiards and Teichmüller curves." Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.3.

Cf. A079896, A361165, A361166, A361168.

A257005 Irregular triangle read by rows: period lengths of periods of Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).

Original entry on oeis.org

1, 2, 2, 1, 3, 5, 4, 1, 3, 1, 4, 2, 5, 2, 5, 4, 2, 1, 6, 4, 7, 6, 4, 11, 6, 3, 5, 1, 1, 6, 2, 2, 1, 10, 3, 7, 8, 2, 9, 7, 6, 3, 2, 1, 11, 9, 7, 8, 5, 8, 2, 8, 4, 2, 21, 10, 7, 7, 1, 8, 4, 2, 1, 10, 4, 3, 1, 9, 5, 12, 6

Offset: 1

Barry R. Smith, Apr 19 2015

The possible positive nonsquare discriminants of binary quadratic forms are given in A079896.
For the definition of Zagier-reduced binary quadratic forms, see A257003.
The row sums give A257003(n), the number of Zagier-reduced forms of discriminant D(n).
The number of entries in row n is A256945(n), the class number of primitive forms of discriminant D(n).

			The table a(n,k) begins:
n/k  1   2   3   4  ...   D(n)    A256945(n)   A257003(n)
1:   1                      5        1             1
2:   2                      8        1             2
3:   2   1                 12        2             3
4:   3                     13        1             3
5:   5                     17        1             5
6:   4   1                 20        2             5
7:   3   1                 21        2             4
8:   4   2                 24        2             6
9:   5   2                 28        2             7
10:  5                     29        1             5
11:  4   2   1             32        3             7
12:  6   4                 33        2            10
13:  7                     37        1             7
14:  6   4                 40        2            10
15: 11                     41        1            11
16:  6   3                 44        2             9
17:  5   1   1             45        3             7
18:  6   2   2   1         48        4            11
19: 10   3                 52        2            13

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A257003, A257006, A256945, A225953, A079896.

a(n,k), n >= 1, k = 1, 2, ..., A256945(n), is the length of the k-th period of the Zagier-reduced forms for discriminant D(n) = A079896(n). The lengths in row n are organized in nonincreasing order.

Offset corrected by Robin Visser, Jun 08 2025

A257006 Irregular triangle read by rows: period lengths of periods of primitive Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).

Original entry on oeis.org

1, 2, 2, 1, 3, 5, 4, 3, 1, 4, 2, 5, 2, 5, 4, 1, 6, 4, 7, 6, 4, 11, 6, 3, 5, 1, 6, 2, 10, 7, 8, 2, 9, 7, 6, 3, 2, 1, 11, 9, 7, 8, 8, 2, 8, 4, 21, 10, 7, 7, 1, 8, 2, 10, 4, 9, 5, 12, 6

Offset: 1

Barry R. Smith, Apr 20 2015

The possible positive nonsquare discriminants of binary quadratic forms are given in A079896.
For the definition of Zagier-reduced binary quadratic forms, see A257003.
A form is primitive if its coefficients are relatively prime.
The row sums give A257004(n), the number of primitive Zagier-reduced forms of discriminant D(n).
The number of entries in row n is A087048(n), the class number of primitive forms of discriminant D(n).

			The table a(n,k) begins:
n/k  1   2   ...   D(n)    A087048(n)   A257004(n)
1:   1               5        1             1
2:   2               8        1             2
3:   2   1          12        2             3
4:   3              13        1             3
5:   5              17        1             5
6:   4              20        1             4
7:   3   1          21        2             4
8:   4   2          24        2             6
9:   5   2          28        2             7
10:  5              29        1             5
11:  4   1          32        2             5
12:  6   4          33        2            10
13:  7              37        1             7
14:  6   4          40        2            10
15: 11              41        1            11
16:  6   3          44        2             9
17:  5   1          45        2             6
18:  6   2          48        2             8
19: 10              52        1            10

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A257004, A257005, A079896, A225953, A079896.

a(n,k), n >= 1, k = 1, 2, ..., A079896(n), is the length of the k-th period of the primitive Zagier-reduced forms of discriminant D(n) = A079896(n). The lengths in row n are organized in nonincreasing order.

Offset corrected by Robin Visser, Jun 08 2025

cp's OEIS Frontend

A361166 Number of elliptic points of order 2 in Weierstrass curve with discriminant A079896(n) in moduli space M_4 of compact Riemann surfaces of genus 4.

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A361168 Number of cusps in Weierstrass curve with discriminant A079896(n) in moduli space M_4 of compact Riemann surfaces of genus 4.

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A226693 Squarefree parts of A079896(n), n>= 1.

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A267857 Length of the period of the continued fraction for the square root of D, the discriminant of indefinite binary quadratic forms. D is given in A079896.

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A361157 Genus of Weierstrass curve with discriminant A079896(n) in moduli space M_2 of compact Riemann surfaces of genus 2.

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A361158 Number of elliptic points of order 2 in Weierstrass curve with discriminant A079896(n) in moduli space M_2 of compact Riemann surfaces of genus 2.

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A361159 Number of cusps in Weierstrass curve with discriminant A079896(n) in moduli space M_2 of compact Riemann surfaces of genus 2.

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A361167 Number of elliptic points of order 3 in Weierstrass curve with discriminant A079896(n) in moduli space M_4 of compact Riemann surfaces of genus 4.

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A257005 Irregular triangle read by rows: period lengths of periods of Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).

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A257006 Irregular triangle read by rows: period lengths of periods of primitive Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).

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