Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #20 Aug 12 2022 09:23:51
%S 1,1,1,1,3,2,1,1,4,3,3,3,3,3,3,1,5,2,3,3,4,3,5,3,3,4,5,3,4,4,4,1,4,4,
%T 3,4,4,3,3,5,4,5,3,3,4,4,5,4,6,4,5,4,5,4,5,4,4,4,5,5,4,5,5,1,4,4,5,3,
%U 4,5,5,4,4,7,4,4,4,5,5,5,4,4,4,4,4,5,4
%N The maximum number of times one reaches a single position during the grasshopper procedure.
%C The grasshopper procedure: n positions are evenly spaced around a circle, a grasshopper hops randomly to any position, after the k-th hop, the grasshopper looks clockwise and counterclockwise k positions. If one of the positions has been visited less often then the other, it hops there; if both positions have been visited an equal number of times, it hops k steps in the clockwise position. (See Mathematics Stack Exchange link for more details.)
%C a(n) >= (A329230(n)-1)/(n-1).
%C Least values of n such that a(n) = 1, 2, 3, etc are 1, 6, 5, 9, 17, 49, 74, 198, 688, 1745 etc.
%C Conjecture: a(n) = 1 if and only if n = 3, n = 7, or n = 2^k for some k.
%C Conjecture: The largest values of n for which a(n) = 2, 3, 4, 5 respectively are n = 18, 68, 381, 1972.
%C If the second conjecture is true, then 2, 3, 4, and 5 appear 2, 19, 87, and 313 times respectively.
%C Conjecture: Every integer greater than 1 appears in this sequence a finite number of times.
%H Peter Kagey, <a href="/A329231/b329231.txt">Table of n, a(n) for n = 1..2048</a>
%H Mathematics Stack Exchange User Vepir, <a href="https://math.stackexchange.com/q/3418970/121988">Grasshopper jumping on circles</a>
%Y Cf. A282442, A329230, A329232, A329233.
%K nonn,walk
%O 1,5
%A _Peter Kagey_, Nov 08 2019