A236213 Number of units in the imaginary quadratic field Q(sqrt(-d)), where d > 0 is the n-th squarefree number.

4, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Jonathan Sondow, Jan 29 2014

Keywords

• nonn
• easy

Comments

a(n) = 2 for all n > 3.

Decimal expansion of 959/225. - Elmo R. Oliveira, May 05 2024

Examples

			Q(sqrt(-1)) = Q(i) has units +/-1, +/-i, so a(1) = 4.
Q(sqrt(-3)) has units +/-1, +/-ω, +/-ω^2, where ω = (1 + sqrt(-3))/2, so a(3) = 6.
Q(sqrt(-d)) has units +/-1 for all other squarefree d > 0, so a(n) = 2 for n = 2 and n > 3.
		

References

• Saban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): p. 98, Theorem 5.4.3.
• Ivan Niven & Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 4th Ed. New York: John Wiley & Sons (1980): p. 249, Theorem 9.22.

Links

• M. Hazewinkel, Quadratic field, Encyclopedia of Mathematics, Springer, 2001.
• Eric Weisstein's World of Mathematics, Unit
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A005117, A092205.

Programs

Mathematica

CoefficientList[Series[2 x (2 - x + 2 x^2 - 2 x^3)/(1 - x), {x, 0, 105}], x] (* Michael De Vlieger, Mar 30 2016 *)

Formula

a(n) = A092205(A005117(n)).

G.f.: 2*x*(2 - x + 2*x^2 - 2*x^3)/(1 - x). [Bruno Berselli, Jan 30 2014]