A270352 Denominators of r-Egyptian fraction expansion for 1/Pi, where r = (1, 1/2, 1/4, 1/8, ...)

4, 8, 44, 977, 498723, 138012074956, 45087947486104434546449, 2223745971024423874814212532278502253766982404, 3439676840537267257806008796995789895364959784333600339427716437786254731225969490712842205

Offset: 1

Author

Clark Kimberling, Mar 17 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

			1/Pi = 1/4 + 1/(2*8) + 1/(4*44) + ...
		

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_] := 2/2^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 = 1/Pi; Table[n[x, k], {k, 1, z}]

PARI

r(k) = 2/2^k;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=1/Pi) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 18 2016