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: 1

Author

Barry R. Smith, Apr 19 2015

Keywords

• nonn
• tabf

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).

Examples

			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
		

References

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

Crossrefs

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

Formula

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.

Extensions

Offset corrected by Robin Visser, Jun 08 2025