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Denominators of signed Egyptian fractions 1/(1+a(n)) with sums converging to sqrt(2).
1

%I #4 May 30 2013 16:58:06

%S 5,26,798,1036219,1679769509872,6521391013068322601901238,

%T 285305237462037970325184857571103801772580723343165

%N Denominators of signed Egyptian fractions 1/(1+a(n)) with sums converging to sqrt(2).

%C The algorithm at A226049, with r = sqrt(2), f(n) = 1/(n+1), gives

%C 1 + 1/2 + ... + 1/a(1) - 1/a(2) + 1/a(3) - , ... converging to sqrt(2). The sum 1 + 1/2 + ... - 1/a(12) differs from sqrt(2) by less than 10^(-3200).

%e Sum of the first 7 signed Egyptian fractions: 1/2 + 1/3 + 1/5 + 1/6 - 1/27 + 1/799 - 1/1036229, showing denominators (beginning at 6), a(1)+1, a(2)+1, a(3)+1, ...

%t $MaxExtraPrecision = Infinity; z = 12; f[n_] := 1/(n + 1); g[n_] := (1 - n)/n; r = Sqrt[2]; s = 0; a[1] = NestWhile[# + 1 &, 1, ! (s += f[#]) > r &]; p = Sum[f[n], {n, 1, a[1]}]; a[2] = Floor[g[p - r]]; a[n_] :=

%t Floor[g[((-1)^n) (p - r - Sum[((-1)^k) f[a[k]], {k, 2, n - 1}])]]; Table[a[k], {k, 1, z}]

%Y Cf. A226049, A226052, A226126.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 27 2013