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A306978
Numbers with record number of iterations of x -> A306938(x) required to reach 1 (A306944).
0
1, 2, 4, 7, 21, 49, 85, 253, 442, 766, 1327, 2299, 3982, 11839, 20506, 35518, 61519, 184557, 553645, 966928, 1674769, 2900785, 8701141, 25877593, 44821306, 77676682, 134539960, 402368674, 696922987, 1207106023, 2090768962, 3632578906
OFFSET
1,2
COMMENTS
The record numbers of iterations are 0, 2, 4, 6, 7, 8, 10, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53.
Conjecture: sqrt(3) < a(n)/a(n-1) <= 3. - Daniel Suteu, Mar 20 2019
LINKS
H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example III.3.
EXAMPLE
2 reaches 1 after 2 iterations: 2 -> 3 -> 1;
4 reaches 1 after a record number of 4 iterations: 4 -> 6 -> 2 -> 3 -> 1.
MATHEMATICA
f[n_] := If[Divisible[n, 3], n/3, Floor[n*Sqrt[3]]]; itnum[n_] := Module[{k = n, inc = False, c = 0}, While[k > 1, kk = f[k]; If[inc && kk > k, c = -1; Break[]]; inc = kk > k; k = kk; c++]; c]; s={}; im=-1; Do[i=itnum[n]; If[i>im, im=i; AppendTo[s, n]], {n, 1, 10000}]; s
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Mar 18 2019
STATUS
approved