A393497 Let K be the real quadratic field of discriminant A003658(n). Then a(n) is the index in O_K of the quadratic order generated by units in O_K with norm 1.

1, 2, 1, 3, 16, 1, 2, 3, 5, 8, 24, 6, 640, 3, 7, 4, 40, 1, 195, 32, 3, 534000, 39, 1, 9, 42, 106000, 5, 3, 12754704, 40, 10, 8, 6525, 226592, 2, 273, 2968, 15, 6, 1039424, 1, 16, 48, 305, 6, 4, 3621, 1856, 1, 2, 531, 13, 9384, 126585, 3588, 1360, 7, 896073208080, 56

Offset

2,2

Comments

Let epsilon be the fundamental unit of K, then the quadratic order is Z[epsilon] if epsilon has norm 1, and Z[epsilon^2] is epsilon has norm -1.

If (x0,y0) is the smallest positive solution to x^2 - D*y^2 = 4, D = A003658(n), then a(n) = y0.

Links

Jianing Song, Table of n, a(n) for n = 2..10000

Example

For D = A003658(3) = 8, we have O_K = Z[sqrt(2)], and the fundamental unit epsilon = 1+sqrt(2) has norm -1. The order generated by units of norm 1 is Z[epsilon^2] = Z[3+2*sqrt(2)] = Z[2*sqrt(2)], so the index is 2.
For D = A003658(4) = 12, we have O_K = Z[sqrt(3)], and the fundamental unit epsilon = 2+sqrt(3) has norm 1. The order generated by units of norm 1 is Z[epsilon] = Z[sqrt(3)], so the index is 1.
For D = A003658(5) = 13, we have O_K = Z[(1+sqrt(13))/2], and the fundamental unit epsilon = (3+sqrt(13))/2 has norm -1. The order generated by units of norm 1 is Z[epsilon^2] = Z[(11+3*sqrt(13))/2] = Z[(1+3*sqrt(13))/2], so the index is 3.

Prog

(PARI) b(D) = my(w=quadunit(D)); if(norm(w)==-1, imag(w^2), imag(w))
for(D=2, 200, if(isfundamental(D), print1(b(D), ", ")))

Crossrefs

Cf. A003658, A014046 (index of the quadratic order generated by all units).

Sequence in context: A389129 A153189 A362272 * A095852 A283748 A209365

Adjacent sequences: A393494 A393495 A393496 * A393498 A393499 A393500

Keyword

nonn

Author

Jianing Song, Mar 22 2026

Status

approved