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URL: https://oeis.org/A387618

⇱ A387618 - OEIS

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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.
2
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
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
KEYWORD
nonn
AUTHOR
Robin Visser, Sep 03 2025
STATUS
approved