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A091833 Pierce expansion of 1/zeta(2). 1

%I

%S 2,4,7,22,29,51,173,210,262,417,746,12341,207220,498538,1286415,

%T 2351289,3702952,7664494,54693034,75971438,269954954,6674693008,

%U 13449203581,59799655308,98912303039,948887634688,3557757020909,5898230078743

%N Pierce expansion of 1/zeta(2).

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

%H G. C. Greubel, <a href="/A091833/b091833.txt">Table of n, a(n) for n = 1..1000</a>

%H P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

%H Vlado Keselj, <a href="https://cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf">Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations </a>.

%H Jeffrey Shallit, <a href="http://www.fq.math.ca/Scanned/22-4/shallit1.pdf">Some predictable Pierce expansions</a>, Fib. Quart., 22 (1984), 332-335.

%H Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, <a href="http://dx.doi.org/10.2139/ssrn.145561">On a problem of Alfred Renyi</a>, Economics Working Paper No. 340.

%F 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)).

%F 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)...

%F limit n ->infty a(n)^(1/n) = e.

%t 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 *)

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

%Y Cf. A006275, A006276, A006283.

%Y Cf. A006784 (Pierce expansion definition), A059186.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Mar 09 2004

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Last modified March 25 17:31 EDT 2017. Contains 284082 sequences.