A078878 - OEIS

%I #7 Apr 30 2014 01:38:50

%S 2,6,12,20,36,60,88,132,196,280,396,552,748,1020,1368,1800,2376,3096,

%T 4004,5160,6600,8372,10584,13320,16652,20760,25764,31824,39204,48120,

%U 58800,71688,87120,105520,127512,153640,184628,221364,264792,315920

%N Irreducible polynomial coefficient of singular value associated with sqrt(2n).

%C "We conclude this section by touching on the matter of singular values, k_p, which for us are defined to be the solutions in (0,1) of W_p(k'^2,k^2) = 0. These are often called singular moduli for the function lambda. ... Then, since K'(k)/K(k) is isotone, Theorem 2.3(b) shows that this is the unique solution to K'/K(k_p) = root(p) 0 < k_p < 1.

%C "In the notation of equation (3.2.1), k_p = lambda^*(p) and k'_p=lambda^*(1/p), so that k_p=k(e^{-pi sqrt(p)}) and l_p := k'_p=k(e^{pi/sqrt(p)}). Sophisticated number-theoretic techniques are available for computing k_p for large p, without knowledge of W_p. ... For small p one can solve directly for k_p. Thus k_1 = 1/root(2), k_2 = root(2)-1, k_3 = root(2)(root(3)-1)/4, k_4 = 3-2root(2), ..." - Borwein and Borwein

%C (K'(x)/K(x))^2 = 2n for some x, whose irreducible polynomial is 1-a(n)x+...

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 139.

%e K'/K = sqrt(6) for k a root of 0 = 1 - 12*x + 2*x^2 + 12*x^3 + x^4 so a(3) = 12.

%K nonn

%O 1,1

%A _Michael Somos_, Dec 13 2002