A270586 Denominators of r-Egyptian fraction expansion for Pi - 3, where r(k) = 1/(k+1).

4, 21, 348, 160397, 27477992122, 8361728546142791039570, 65449866064796651219032701504776304475204846, 4242994078960802485293647297249599708082797742348261121304757890775884278785179376866443

Offset

1,1

Comments

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 1..11

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

Pi - 3 = 1/(2*4) + 1/(3*21) + 1/(4*348) + 1/(5*160397) + ...

Mathematica

r[k_] := 1/(k+1); f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Pi - 3; Table[n[x, k], {k, 1, z}]

Crossrefs

Cf. A269993.

Sequence in context: A270482 A126458 A319363 * A048164 A118909 A225157

Adjacent sequences: A270583 A270584 A270585 * A270587 A270588 A270589

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 03 2016

Status

approved