A269999 Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1,1/2,1/3,1/4,...)

8, 31, 719, 17276711, 557951558165893, 1713250424923433306065171045669, 3960162768997467999491098138568123635738830147395528618636887, 148114266323338300606167235125265318767829304330791212171374192569332869541220746054882408155611146661783688512870116687748

Offset: 1

Author

Clark Kimberling, Mar 15 2016

Keywords

• nonn
• frac
• easy

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.

Examples

			Pi - 3 = 1/8 + 1/(2*31) + 1/(3*719) + ...
		

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

Crossrefs

Cf. A269993.

Programs

Mathematica

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

PARI

r(k) = 1/k;
x = Pi - 3;
f(x, k) = if(k<1, x, f(x, k - 1) - r(k)/n(x, k));
n(x, k) = ceil(r(k)/f(x, k - 1));
for(k = 1, 8, print1(n(x, k), ", ")) \\ Indranil Ghosh, Mar 29 2017