A270554 - OEIS

%I #10 Feb 24 2018 14:02:56

%S 2,2,4,89,11084,101449736,10283734953162540,

%T 146727957364669007427252221319539,

%U 22046450568037230736928892396703267030745801074192006421746544107

%N Denominators of r-Egyptian fraction expansion for e - 2, where r(k) = 1/(2k-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="/A270554/b270554.txt">Table of n, a(n) for n = 1..12</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 e - 2 = 1/(1*2) + 1/(3*2) + 1/(5*4) + 1/(7*89) + ...

%t r[k_] := 1/(2k-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 = E - 2; Table[n[x, k], {k, 1, z}]

%o (PARI) r(k) = 1/(2*k-1);

%o f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););

%o a(k, x=exp(1)-2) = ceil(r(k)/f(k-1, x)); \\ _Michel Marcus_, Apr 03 2016

%Y Cf. A269993, A005408, A001113.

%K nonn,frac,easy

%O 1,1

%A _Clark Kimberling_, Apr 02 2016