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A270590 Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (EulerGamma), where r(k) = 1/(k+1). 1

%I #11 Feb 24 2018 17:59:26

%S 1,5,24,1512,2953202,23271987449429,695674431250515976519182860,

%T 5836770961962275507879845242138280180068525903243835399,

%U 252105542081571496083070310148809536964991357782450789266426812999313566935591332769492328764874678005359339022

%N Denominators of r-Egyptian fraction expansion for the Euler-Mascheroni constant (EulerGamma), where r(k) = 1/(k+1).

%C 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.

%C See A269993 for a guide to related sequences.

%H Clark Kimberling, <a href="/A270590/b270590.txt">Table of n, a(n) for n = 1..11</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%e Euler-Mascheroni constant = 1/(2*1) + 1/(3*5) + 1/(4*24) + 1/(5*1512) + ...

%t r[k_] := 1/(k+1); f[x_, 0] = x; z = 10;

%t n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]

%t f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]

%t x = EulerGamma; Table[n[x, k], {k, 1, z}]

%Y Cf. A269993.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Apr 04 2016

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Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)