A270552 Denominators of r-Egyptian fraction expansion for Pi - 3, where r(k) = 1/(2k-1).

8, 21, 278, 669885, 883049829180, 1070939942519425635457275, 8036017127630347959082917393914880002819233759027, 144463610576667598395827626720494192280404388648949928084764924235587554966022803149344499311620245

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/(1*8) + 1/(3*21) + 1/(5*278) + 1/(7*669885) + ...

Mathematica

r[k_] := 1/(2k-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}]

Prog

(PARI) r(k) = 1/(2*k-1);
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=Pi-3) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

Crossrefs

Cf. A269993, A005408, A000796.

Sequence in context: A231525 A228756 A228504 * A156239 A141369 A060390

Adjacent sequences: A270549 A270550 A270551 * A270553 A270554 A270555

Keyword

nonn,frac,easy

Author

Clark Kimberling, Apr 02 2016

Status

approved