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A226051 Denominators of signed Egyptian fractions with sums converging to pi. 2
13, 25, 685, 1046508, 3249663242463, 11242879629599233747822858, 296966610697839275946742496484126789076162611540968 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Using the algorithm defined at A226049 with r = pi and f(n) = 1/n gives

r = sum{1/k, k=1..13} - 1/25 + 1/685 - 1/1046508 + ..., of which the 22nd partial sum differs from the pi by less than 10^(-800).  For a guide to related sequences, see A226049.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..12

EXAMPLE

1 + 1/2 + ... + 1/12 < pi < 1 + 1/2 + ... + 1/13, so a(1) = 13.

1 + 1/2 + ... + 1/13 - 1/25 < pi, so a(2) = 25.

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

MATHEMATICA

$MaxExtraPrecision = Infinity;

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}]]

CROSSREFS

Cf. A226049.

Sequence in context: A122003 A123827 A105796 * A238338 A040156 A296752

Adjacent sequences:  A226048 A226049 A226050 * A226052 A226053 A226054

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 24 2013

STATUS

approved

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Last modified September 17 19:12 EDT 2019. Contains 327137 sequences. (Running on oeis4.)