A091833 Pierce expansion of 1/zeta(2).

1, 2, 4, 7, 22, 29, 51, 173, 210, 262, 417, 746, 12341, 207220, 498538, 1286415, 2351289, 3702952, 7664494, 54693034, 75971438, 269954954, 6674693008, 13449203581, 59799655308, 98912303039, 948887634688, 3557757020909, 5898230078743

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..1001 [a(1)=1 inserted by Georg Fischer, Nov 20 2020]

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.

Formula

Let u(0) = Pi^2/6 and u(n+1) = u(n)/frac(u(n)) where frac(x) is the fractional part of x; then a(n) = floor(u(n)).

1/zeta(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->oo} 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[1/Zeta[2], 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

Prog

(PARI) default(realprecision, 100000); r=zeta(2); for(n=1, 100, s=(r/(r-floor(r))); print1(floor(r), ", "); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]

Crossrefs

Cf. A006275, A006276, A006283.

Cf. A006784 (Pierce expansion definition), A059186.

Sequence in context: A153550 A102984 A103017 * A171978 A290571 A026080

Adjacent sequences: A091830 A091831 A091832 * A091834 A091835 A091836

Keyword

nonn

Author

Benoit Cloitre, Mar 09 2004

Extensions

a(1)=1 inserted by Georg Fischer, Nov 20 2020

Status

approved