A084540 a(n) is the algebraic order of the elliptic lambda function lambda^*(n), where lambda^*(n) is the value of k_n such that K'(k_n)/K(k_n) = sqrt(n), K(k) is a complete elliptic function and K'(k) is the complementary function.

2, 2, 4, 2, 8, 4, 4, 4, 8, 4, 12, 4, 8, 8, 8, 4, 16, 4, 12, 8, 16, 4, 12, 8, 8, 12, 12, 4, 24, 8, 12, 8, 16, 8, 24, 8, 8, 12, 16, 8, 32, 8, 12, 12, 16, 8, 20, 8, 16, 12, 24, 8, 24, 12, 16, 16, 16, 4, 36, 8, 24, 16, 16, 8, 32, 16, 12, 16, 32, 8, 28, 8, 16, 20, 24, 12, 32, 8, 20, 16, 24, 8, 36, 16, 16, 20, 24, 8, 48, 16, 24, 12, 16, 16, 32, 16, 16, 16, 24, 8

Offset

1,1

Comments

If we consider the polynomial (1-lambda(t)+lambda(t)^2)^3-(j(t)/256)*lambda(t)^2*(1-lambda(t))^2 = 0 for D <> 5 (mod 8) where D is the fundamental discriminant of the imaginary quadratic field, then this polynomial factorizes into a product of three quadratic polynomials over Q(j(t)). From this follows that 2*A000003(n) gives us the highest exponent of a minimal polynomial with a root in lambda(I*sqrt(n)). This leads to the relation that a(n) is also given by 2*gcd(2, n+1)*A000003(n) for n > 1 because lambda^* is related to lambda by square root. By using the substitution lambda(I*sqrt(4*n)) = ((1-sqrt(1-lambda(I*sqrt(n))^2))/(1+sqrt(1-lambda(I*sqrt(n))^2)))^2 a somehow nicer expression which holds for all n can be derived: gcd(2, n+1)*A000003(4*n). - Thomas Scheuerle, Oct 30 2025

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Formula

a(n) = gcd(2, n+1)*A000003(4*n). - Thomas Scheuerle, Oct 30 2025

Example

k_3 = (1/4)*sqrt(2)*(sqrt(3)-1) this is a root of 1 - 16*x^2 + 16*x^4 (row 3 from A389848), so a(3) = 4.

Mathematica

Table[Exponent[MinimalPolynomial[RootApproximant[N[Sqrt[ModularLambda[I*Sqrt[n]]], 100*n]], x], x], {n, 1, 50}] (* Vaclav Kotesovec, Apr 07 2018 *)

Prog

(PARI)
a(n) = poldegree(factor(numerator(subst(polclass(-(n*4)), 'x, 256*(1-x^2+x^4)^3/(x^4*(1-x^2)^2))))[, 1][1]); \\ Thomas Scheuerle, Oct 30 2025
a(n) = qfbclassno(-16*n)*gcd(2, n+1); \\ Thomas Scheuerle, Oct 30 2025

Crossrefs

Cf. A000003, A115977, A389848 (corresponding minimal polynomials).

Sequence in context: A270365 A200147 A235063 * A364557 A297112 A259192

Adjacent sequences: A084537 A084538 A084539 * A084541 A084542 A084543

Keyword

nonn

Author

Eric W. Weisstein, May 29 2003

Extensions

a(29)-a(100) from Vaclav Kotesovec, Apr 07 2018

Status

approved