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 A270518 Denominators of r-Egyptian fraction expansion for sqrt(1/3), where r(k) = 1/k!. 1
 2, 7, 29, 239, 35642, 4939700112, 48108453420633293272, 444429875521548685791697227054499321900, 25562938514216590071082104331351977875333056562865491878765431482309855946304 (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..12 Eric Weisstein's World of Mathematics, Egyptian Fraction EXAMPLE sqrt(1/3) = 1/(1*2) + 1/(2*7) + 1/(6*29) + 1/(24*239) + ... 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[1/3]; 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(1/3)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016 CROSSREFS Cf. A269993, A000142, A020760. Sequence in context: A346427 A003437 A192410 * A094475 A093034 A125174 Adjacent sequences: A270515 A270516 A270517 * A270519 A270520 A270521 KEYWORD nonn,frac,easy AUTHOR Clark Kimberling, Mar 30 2016 STATUS approved

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Last modified January 28 20:13 EST 2023. Contains 359905 sequences. (Running on oeis4.)