This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091833 #36 Jan 05 2025 19:51:37
%S A091833 1,2,4,7,22,29,51,173,210,262,417,746,12341,207220,498538,1286415,
%T A091833 2351289,3702952,7664494,54693034,75971438,269954954,6674693008,
%U A091833 13449203581,59799655308,98912303039,948887634688,3557757020909,5898230078743
%N A091833 Pierce expansion of 1/zeta(2).
%C A091833 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 A091833 G. C. Greubel, <a href="/A091833/b091833.txt">Table of n, a(n) for n = 1..1001</a> [a(1)=1 inserted by Georg Fischer, Nov 20 2020]
%H A091833 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 A091833 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 A091833 Jeffrey Shallit, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/22-4/shallit1.pdf">Some predictable Pierce expansions</a>, Fib. Quart., 22 (1984), 332-335.
%H A091833 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 A091833 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 A091833 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 A091833 Limit_{n->oo} a(n)^(1/n) = e.
%t A091833 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 A091833 (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]
%Y A091833 Cf. A006275, A006276, A006283.
%Y A091833 Cf. A006784 (Pierce expansion definition), A059186.
%K A091833 nonn
%O A091833 1,2
%A A091833 _Benoit Cloitre_, Mar 09 2004
%E A091833 a(1)=1 inserted by _Georg Fischer_, Nov 20 2020