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

1,1

Comments

a(n) = 2 for all 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

Table of n, a(n) for n=1..105.

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).

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]

Example

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.

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 *)

Crossrefs

Cf. A005117, A092205.

Sequence in context: A019719 A205111 A340177 * A016694 A175038 A035505

Adjacent sequences: A236210 A236211 A236212 * A236214 A236215 A236216

Keyword

nonn,easy

Author

Jonathan Sondow, Jan 29 2014

Status

approved