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

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

1,2

Comments

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).

References

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

Links

Table of n, a(n) for n=1..59.

Formula

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.

Example

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

Crossrefs

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

Sequence in context: A368338 A344583 A349414 * A363565 A139687 A188181

Adjacent sequences: A257003 A257004 A257005 * A257007 A257008 A257009

Keyword

nonn,tabf

Author

Barry R. Smith, Apr 20 2015

Extensions

Offset corrected by Robin Visser, Jun 08 2025

Status

approved