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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.
0
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 30 2010
STATUS
approved