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 A270520 Denominators of r-Egyptian fraction expansion for sqrt(3) - 1, where r(k) = 1/k!. 1
 2, 3, 3, 5, 6, 14, 28, 352, 18574, 44518346, 400826881311158, 25342673472297507115832358714, 62130292590921086469117151395751018383242940308998211770 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1).  Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k).  Then x = r(1)/n(1)) + r(2)/(n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x. See A269993 for a guide to related sequences. LINKS Clark Kimberling, Table of n, a(n) for n = 1..20 Eric Weisstein's World of Mathematics, Egyptian Fraction EXAMPLE sqrt(3) - 1 = 1/(1*2) + 1/(2*3) + 1/(6*3) + 1/(24*5) + ... MATHEMATICA r[k_] := 1/k!; f[x_, 0] = x; z = 10; n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]] f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k] x = Sqrt - 1; Table[n[x, k], {k, 1, z}] PROG (PARI) r(k) = 1/k!; f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); ); a(k, x=sqrt(3)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016 CROSSREFS Cf. A269993, A000142, A160390. Sequence in context: A238003 A218932 A056878 * A092557 A333295 A144680 Adjacent sequences:  A270517 A270518 A270519 * A270521 A270522 A270523 KEYWORD nonn,frac,easy AUTHOR Clark Kimberling, Mar 30 2016 STATUS approved

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Last modified April 12 18:47 EDT 2021. Contains 342931 sequences. (Running on oeis4.)