A270482 Denominators of r-Egyptian fraction expansion for Pi - 3, where r(k) = 1/Prime(k).

4, 21, 278, 669885, 722495314784, 518991396646391611675178, 1140942688502784034709261534054801143469100357774, 3057883757819924789715910581052335037849699410792383985720220681289207821611036680026225078199178

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/(5*278) + 1/(7*669885) + ...

Mathematica

r[k_] := 1/Prime[k]; 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 - 3i; Table[n[x, k], {k, 1, z}]

Crossrefs

Cf. A269993, A000040.

Sequence in context: A165627 A324954 A198050 * A126458 A319363 A270586

Adjacent sequences: A270479 A270480 A270481 * A270483 A270484 A270485

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 30 2016

Status

approved