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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).

%I A257005 #26 Jun 09 2025 00:53:58
%S A257005 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,
%T A257005 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,
%U A257005 3,1,9,5,12,6
%N A257005 Irregular triangle read by rows: period lengths of periods of Zagier-reduced binary quadratic forms with discriminants D(n) = A079896(n).
%C A257005 The possible positive nonsquare discriminants of binary quadratic forms are given in A079896.
%C A257005 For the definition of Zagier-reduced binary quadratic forms, see A257003.
%C A257005 The row sums give A257003(n), the number of Zagier-reduced forms of discriminant D(n).
%C A257005 The number of entries in row n is A256945(n), the class number of primitive forms of discriminant D(n).
%D A257005 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%F A257005 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.
%e A257005 The table a(n,k) begins:
%e A257005 n/k  1   2   3   4  ...   D(n)    A256945(n)   A257003(n)
%e A257005 1:   1                      5        1             1
%e A257005 2:   2                      8        1             2
%e A257005 3:   2   1                 12        2             3
%e A257005 4:   3                     13        1             3
%e A257005 5:   5                     17        1             5
%e A257005 6:   4   1                 20        2             5
%e A257005 7:   3   1                 21        2             4
%e A257005 8:   4   2                 24        2             6
%e A257005 9:   5   2                 28        2             7
%e A257005 10:  5                     29        1             5
%e A257005 11:  4   2   1             32        3             7
%e A257005 12:  6   4                 33        2            10
%e A257005 13:  7                     37        1             7
%e A257005 14:  6   4                 40        2            10
%e A257005 15: 11                     41        1            11
%e A257005 16:  6   3                 44        2             9
%e A257005 17:  5   1   1             45        3             7
%e A257005 18:  6   2   2   1         48        4            11
%e A257005 19: 10   3                 52        2            13
%Y A257005 Cf. A257003, A257006, A256945, A225953, A079896.
%K A257005 nonn,tabf
%O A257005 1,2
%A A257005 _Barry R. Smith_, Apr 19 2015
%E A257005 Offset corrected by _Robin Visser_, Jun 08 2025