A268058 Maximum value of n-th row of A268057.
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, 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, 7, 7, 6, 4, 6, 6, 5, 7, 7, 6, 8, 5, 5, 6, 9, 5, 6, 6, 7
Offset: 1
Keywords
Links
- Peter Kagey, Table of n, a(n) for n = 1..10000
- Zachary Chase and Mayank Pandey, On the length of Pierce expansions, arXiv preprint (2022). arXiv:2211.08374 [math.NT]
- P. Erdős and J. O. Shallit, New bounds on the length of finite Pierce and Engel series, Journal de Théorie des Nombres de Bordeaux 3:1 (1991), pp. 43-53.
- Vlado Kešelj, Length of finite Pierce series: theoretical analysis and numerical calculations (1996), 27 pp.
- J. O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart. 24 (1986), pp. 22-40.
- Reddit user zifyoip, First 100 terms.
- Index entries for sequences related to Engel expansions
Programs
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PARI
P(a,b)=my(n); while(b, b=a%b; n++); n a(n)=my(t=1); for(b=2,n-1, t=max(P(n,b),t)); t \\ Charles R Greathouse IV, Nov 26 2016
Formula
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