A387618 Smallest positive integer k such that there exists an algebraic integer in Q(sqrt(D)) of norm -k, where D = A005117(n) is the n-th squarefree number.

1, 2, 1, 2, 3, 1, 2, 1, 5, 6, 1, 2, 3, 2, 7, 1, 1, 5, 3, 2, 9, 10, 1, 2, 3, 1, 6, 2, 5, 11, 2, 1, 6, 2, 1, 2, 1, 13, 1, 2, 2, 5, 5, 7, 1, 1, 7, 14, 15, 1, 2, 1, 2, 6, 1, 3, 3, 5, 14, 1, 1, 2, 3, 5, 1, 2, 1, 10, 3, 1, 2, 10, 2, 19, 1, 2, 3, 2, 1, 2, 3, 2, 1, 17, 2, 5, 21, 22, 1, 2, 1, 3, 6

Offset

2,2

Comments

If D = A005117(n) is 2 or 3 (mod 4), then a(n) is the smallest positive integer k of the form D*X^2 - Y^2 for some integers X and Y. Otherwise if D = A005117(n) is 1 (mod 4), then a(n) is the smallest positive integer k of the form ((D-1)/4)*X^2 - X*Y - Y^2 for some integers X and Y.

Links

Robin Visser, Table of n, a(n) for n = 2..10000

Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.

Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Formula

a(n) = 1 if and only if A005117(n) is a member of the sequence A003654.

a(n) < 2*sqrt(A005117(n)) for all n > 1.

Example

For n = 2, the field Q(sqrt(A005117(2))) = Q(sqrt(2)) contains the integral element 1 + sqrt(2) which has norm -1, thus a(2) = 1.
For n = 3, the field Q(sqrt(A005117(3))) = Q(sqrt(3)) contains the integral element 1 + sqrt(3) which has norm -2, but contains no unit of norm -1, thus a(3) = 2.
For n = 4, the field Q(sqrt(A005117(4))) = Q(sqrt(5)) contains the integral element 1/2 + sqrt(5)/2 which has norm -1, thus a(4) = 1.

Prog

(SageMath)
def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K.<a> = QuadraticField(D); UK = K.unit_group()
    if UK.fundamental_units()[0].norm() == -1: return 1
    idls = K.ideals_of_bdd_norm(D)
    for (k, I) in [(x, y) for x in range(1, D) for y in idls[x]]:
        ans = -(I.gens_reduced()[0].norm())
        if I.is_principal() and ans > 0: return ans

Crossrefs

Cf. A003654, A005117, A387207.

Sequence in context: A183198 A249160 A386641 * A346913 A269970 A333518

Adjacent sequences: A387615 A387616 A387617 * A387619 A387620 A387621

Keyword

nonn

Author

Robin Visser, Sep 03 2025

Status

approved