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%I #8 Feb 23 2018 22:02:46
%S 1,3,24,815,2263886,9073564639850,176228569027146222763928594,
%T 84205747605016031994416006285857418872429042805656089
%N Denominators of signed Egyptian fractions with sums converging to sqrt(1/2).
%C Using the algorithm defined at A226049 with r = sqrt(1/2) and f(n) = 1/n gives r = 1 + 1/2 + 1/3 - 1/24 + 1/815 + 1/2263886 + ..., of which the 14th partial sum differs from the r by less than 10^(-1700). For a guide to related sequences, see A226049.
%H Clark Kimberling, <a href="/A226053/b226053.txt">Table of n, a(n) for n = 1..12</a>
%e Let r = sqrt(1/2). Then
%e 1 + 1/2 < r < 1 + 1/2 + 1/3, so a(1) = 3.
%e 1 + 1/2 + 1/3 -1/24 < r, so a(2) = 24.
%e 1 + 1/2 + 1/3 -1/24 + 1/815 > r, so a(3) = 815.
%t $MaxExtraPrecision = Infinity;
%t nn = 12; f[n_] := 1/n; r = Sqrt[1/2]; s = 0; b[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; u[1] = Sum[f[n], {n, 1, b[1]}]; c[1] = Floor[1/(u[1] - r)]; v[1] = u[1] - 1/c[1]; n = 1; While[n < nn/2, n++; b[n] = Floor[1/(r - v[n - 1])]; u[n] = v[n - 1] + 1/b[n]; c[n] = Floor[1/(u[n] - r)]; v[n] = u[n] - 1/c[n]]; a = Riffle[Table[b[i], {i, 1, nn/2}], Table[c[i], {i, 1, nn/2}]]
%Y Cf. A226049.
%K nonn
%O 1,2
%A _Clark Kimberling_, May 24 2013
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