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

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

0,2

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.

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).

References

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

Links

Table of n, a(n) for n=0..70.

Formula

a(n,k), n >= 0, 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.

Example

The table a(n,k) begins:
n/k  1   2   3   4  ...   D(n)    A256945(n)   A257003(n)
0:   1                      5        1             1
1:   2                      8        1             2
2:   2   1                 12        2             3
3:   3                     13        1             3
4:   5                     17        1             5
5:   4   1                 20        2             5
6:   3   1                 21        2             4
7:   4   2                 24        2             6
8:   5   2                 28        2             7
9:   5                     29        1             5
10:  4   2   1             32        3             7
11:  6   4                 33        2            10
12:  7                     37        1             7
13:  6   4                 40        2            10
14: 11                     41        1            11
15:  6   3                 44        2             9
16:  5   1   1             45        3             7
17:  6   2   2   1         48        4            11
18: 10   3                 52        2            13

Crossrefs

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

Sequence in context: A284644 A079956 A140717 * A160232 A026300 A099514

Adjacent sequences: A257002 A257003 A257004 * A257006 A257007 A257008

Keyword

nonn,tabf

Author

Barry R. Smith, Apr 19 2015

Status

approved