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%I #24 Jan 14 2023 09:11:48
%S 1,1,2,2,3,2,3,3,3,3,5,3,4,4,3,3,5,4,6,4,4,5,6,3,5,4,5,4,5,4,6,4,5,6,
%T 7,4,5,6,5,4,6,4,6,5,5,7,8,4,7,5,6,5,9,5,6,5,6,6,8,4,7,6,6,5,7,5,8,7,
%U 7,7,6,4,6,6,5,7,7,6,8,5,5,6,9,5,6,6,7
%N Maximum value of n-th row of A268057.
%H Peter Kagey, <a href="/A268058/b268058.txt">Table of n, a(n) for n = 1..10000</a>
%H Zachary Chase and Mayank Pandey, <a href="https://arxiv.org/abs/2211.08374">On the length of Pierce expansions</a>, arXiv preprint (2022). arXiv:2211.08374 [math.NT]
%H P. Erdős and J. O. Shallit, <a href="http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1991__3_1/JTNB_1991__3_1_43_0/JTNB_1991__3_1_43_0.pdf">New bounds on the length of finite Pierce and Engel series</a>, Journal de Théorie des Nombres de Bordeaux 3:1 (1991), pp. 43-53.
%H Vlado Kešelj, <a href="https://cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf">Length of finite Pierce series: theoretical analysis and numerical calculations</a> (1996), 27 pp.
%H J. O. Shallit, <a href="https://www.fq.math.ca/Scanned/24-1/shallit.pdf">Metric theory of Pierce expansions</a>, Fibonacci Quart. 24 (1986), pp. 22-40.
%H Reddit user zifyoip, <a href="https://www.reddit.com/r/math/comments/409dfe/does_anyone_know_anything_about_this_idea_i/cysj882">First 100 terms.</a>
%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%F Chase & Pandey prove that a(n) = O(n^e) for any e > 19/59 = 0.322..., improving on Kešelj, Erdős & Shallit, and Shallit. - _Charles R Greathouse IV_, Jan 13 2023
%o (PARI) P(a,b)=my(n); while(b, b=a%b; n++); n
%o a(n)=my(t=1); for(b=2,n-1, t=max(P(n,b),t)); t \\ _Charles R Greathouse IV_, Nov 26 2016
%Y Cf. A268057, A268059, A268060.
%K nonn
%O 1,3
%A _Peter Kagey_, Jan 25 2016
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