A226051 - OEIS

%I #12 Apr 09 2021 19:04:09

%S 13,25,685,1046508,3249663242463,11242879629599233747822858,

%T 296966610697839275946742496484126789076162611540968

%N Denominators of signed Egyptian fractions with sums converging to Pi.

%C Using the algorithm defined at A226049 with r = Pi and f(n) = 1/n gives r = (Sum_{k=1..13} 1/k) - 1/25 + 1/685 - 1/1046508 + ..., of which the 22nd partial sum differs from Pi by less than 10^(-800). For a guide to related sequences, see A226049.

%H Clark Kimberling, <a href="/A226051/b226051.txt">Table of n, a(n) for n = 1..12</a>

%e 1 + 1/2 + ... + 1/12 < Pi < 1 + 1/2 + ... + 1/13, so a(1) = 13.

%e 1 + 1/2 + ... + 1/13 - 1/25 < Pi, so a(2) = 25.

%e 1 + 1/2 + ... + 1/13 - 1/25 + 1/685 > Pi, so a(3) = 685.

%t $MaxExtraPrecision = Infinity;

%t nn = 10; f[n_] := 1/n; r = Pi; 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 1,1

%A _Clark Kimberling_, May 24 2013