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A078878
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Irreducible polynomial coefficient of singular value associated with sqrt(2n).
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0
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2, 6, 12, 20, 36, 60, 88, 132, 196, 280, 396, 552, 748, 1020, 1368, 1800, 2376, 3096, 4004, 5160, 6600, 8372, 10584, 13320, 16652, 20760, 25764, 31824, 39204, 48120, 58800, 71688, 87120, 105520, 127512, 153640, 184628, 221364, 264792, 315920
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| "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.
"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
(K'(x)/K(x))^2 = 2n for some x, whose irreducible polynomial is 1-a(n)x+...
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REFERENCES
| J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 139.
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EXAMPLE
| K'/K = sqrt(6) for k a root of 1-12x+2x^2+12x^3+x^4 so a(3)=12.
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CROSSREFS
| Sequence in context: A184432 A003274 A121315 * A095361 A095362 A099885
Adjacent sequences: A078875 A078876 A078877 * A078879 A078880 A078881
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Dec 13, 2002
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