A220396 - OEIS

%I #14 Jul 26 2021 01:48:21

%S 2,7,18,4,2,2,3,1466,1464,9,24,4,2,9,104,60,8,2,3,6,4,2,2,2,2,2,2,2,2,

%T 2,3,32,30,2,13,36,6,4,3,6,6,4,4,6,2,4,6,2,4,6,9,24,4,5,8,2,2,2,2,2,3,

%U 20

%N A modified Engel expansion of the Euler-Mascheroni constant gamma.

%C See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

%H Peter Bala, <a href="/A220393/a220393.pdf">A modified Engel expansion</a>

%H S. Crowley, <a href="http://arxiv.org/abs/1210.5652">Integral transforms of the harmonic sawtooth map, the Riemann zeta function, fractal strings, and a finite reflection formula</a>, arXiv:1210.5652 [math.NT], 2012-2020.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>

%F Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = gamma (see A001620). Then a(1) = 1 + floor(1/x) and, for n >= 1, a(n+1) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

%F Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion sqrt(2) = Sum_{n>=1} 1/P(n) = 1/2 + 1/(2*7) + 1/(2*7*18) + 1/(2*7*18*4) + 1/(2*7*18*4*2) + .... The error made in truncating this series to n terms is less than the n-th term.

%Y Cf. A001620, A053977, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220397, A220398.

%K nonn,easy

%O 1,1

%A _Peter Bala_, Dec 13 2012