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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.

%I A172259 #25 Oct 22 2023 17:04:00
%S A172259 1,2,5,14,38,101,275,746,2026,5507,14969,40689,110604,300652,817255,
%T A172259 2221528,6038739,16414993,44620576,121291299,329703934,896228212,
%U A172259 2436200862,6622280533,18001224835,48932402358,133012060152,361564266077,982833574297,2671618645410
%N 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.
%C A172259 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.
%D A172259 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.
%D A172259 M. Abramowitz and I. Stegun, "Elliptic Integrals", Chapter 17 of Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).
%D A172259 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.
%H A172259 A. Cayley, <a href="http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=cayl005">An Elementary Treatise on Elliptic Functions</a>, G. Bell and Sons, London, 1895, p. 56.
%H A172259 F. Clarke, <a href="http://www-maths.swan.ac.uk/staff/fwc/Gregynog-slides.pdf">The Taylor Series Coefficients of the Jacobi Elliptic Functions</a>, slides. [broken link]
%F A172259 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.
%e A172259 a(3) = 38 because floor(CK(1/37)) = 4 and floor(CK(1/38)) = 5.
%p A172259 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:
%Y A172259 Cf. elliptic functions: A001936, A002318, A001937, A001934, A001938, A002754, A001939, A001940, A001941, A002753, A006089, A004005.
%K A172259 nonn
%O A172259 1,2
%A A172259 _Michel Lagneau_, Jan 30 2010