A270522 Denominators of r-Egyptian fraction expansion for 1/Pi, where r(k) = 1/k!.

4, 8, 29, 664, 953145, 196766735533, 10443300766535420462458, 262258539032056050210886969489481493453849515, 12566685711857426723146448629806959643446259099771771155403988019749226774526777623471845

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..12

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

Example

1/Pi = 1/(1*4) + 1/(2*8) + 1/(6*29) + 1/(24*664) + ...

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 = 1/Pi; Table[n[x, k], {k, 1, z}]

Prog

(PARI) r(k) = 1/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 31 2016

Crossrefs

Cf. A269993, A000142, A049541.

Sequence in context: A322784 A297638 A256456 * A372384 A317583 A020331

Adjacent sequences: A270519 A270520 A270521 * A270523 A270524 A270525

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 30 2016

Status

approved