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A091833 Pierce expansion of 1/zeta(2). 0


%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

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

%H Author?, <a href="http://www.econ.upf.es/deehome/what/wpapers/postscripts/340.pdf">On a problem of Alfred Renyi </a>

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

%H Jeffrey Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/sppe.ps">Some predictable Pierce expansions</a>, Fib. Quart., 22 (1984), 332-335.

%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

%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 December 18 06:31 EST 2014. Contains 252079 sequences.