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A091846 Pierce expansion of log(2). 1
1, 3, 12, 21, 51, 57, 73, 85, 96, 1388, 4117, 5268, 9842, 11850, 16192, 19667, 29713, 76283, 460550, 1333597, 1462506, 9400189, 13097390, 30254851, 190193800, 201892756, 431766247, 942050077, 6204785761, 16684400052, 23762490104 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If u(0)=exp(1/m) m integer>=1 and u(n+1)=u(n)/frac(u(n)) then floor(u(n))=m*n.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..500

P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .

Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.

Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, On a Problem of Alfred Renyi, Economics Working Paper No. 340.

Eric Weisstein's World of Mathematics, Pierce Expansion

FORMULA

Let u(0)=1/log(2) and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n)).

log(2) = 1/a(1) - 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) - 1/(a(1)*a(2)*a(3)*a(4)) +- ...

limit n-->infinity a(n)^(1/n) = e.

MATHEMATICA

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Log[2], 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

PROG

(PARI) r=1/log(2); for(n=1, 30, r=r/(r-floor(r)); print1(floor(r), ", "))

CROSSREFS

Cf. A006275, A006276, A006283, A006284.

Cf. A006784 (Pierce expansion definition), A059180.

Sequence in context: A210282 A160167 A160412 * A061262 A051656 A074004

Adjacent sequences:  A091843 A091844 A091845 * A091847 A091848 A091849

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Mar 09 2004

STATUS

approved

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Last modified December 7 11:34 EST 2016. Contains 278874 sequences.