A270587 - OEIS

%I #8 Feb 24 2018 14:01:28

%S 2,3,37,17282,292943843,105008217236608424,

%T 15721030895760728441349083553863848,

%U 293537352086847609928310098819595906620056713567902655641220410550838

%N Denominators of r-Egyptian fraction expansion for 1/e, 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="/A270587/b270587.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 1/e = 1/(2*2) + 1/(3*3) + 1/(4*37) + 1/(5*17282) + ...

%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 = 1/E; Table[n[x, k], {k, 1, z}]

%Y Cf. A269993.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Apr 03 2016