A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

1, 2, 5, 14, 38, 101, 275, 746, 2026, 5507, 14969, 40689, 110604, 300652, 817255, 2221528, 6038739, 16414993, 44620576, 121291299, 329703934, 896228212, 2436200862, 6622280533, 18001224835, 48932402358, 133012060152, 361564266077, 982833574297, 2671618645410

Offset

1,2

Comments

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt and the complete elliptic integral CK is defined by CK(k) = F(1,sqrt(1-k^2)). We calculate the values of CK(k) with k = 1/p, p = 1,2,3, ... and we propose a very interesting property: a(n+1)/a(n) tends toward e = 2.7182818... when n tends to infinity. For example, a(8) / a(7) = 2.718281581; a(9) / a(8) = 2.7182817562.

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.

M. Abramowitz and I. Stegun, "Elliptic Integrals", Chapter 17 of Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

Links

Table of n, a(n) for n=1..30.

A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 56.

F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]

Formula

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt. CK is defined by CK(k) = F(1,sqrt(1-k^2)). a(n) is the n-th integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

Example

a(3) = 38 because floor(CK(1/37)) = 4 and floor(CK(1/38)) = 5.

Maple

a0:=1:for p from 1 to 1000 do:a:= evalf(EllipticCK(1/p)):if floor(a)=a0+1 then print(p):a0:=floor(a):else fi:od:

Crossrefs

Cf. elliptic functions: A001936, A002318, A001937, A001934, A001938, A002754, A001939, A001940, A001941, A002753, A006089, A004005.

Sequence in context: A148312 A148313 A228952 * A292327 A084085 A052985

Adjacent sequences: A172256 A172257 A172258 * A172260 A172261 A172262

Keyword

nonn

Author

Michel Lagneau, Jan 30 2010

Status

approved