A258756 - OEIS

%I #12 Feb 22 2021 13:27:14

%S 1,1,2,2,3,3,4,3,5,5,4,4,7,3,4,4,3,3,4,3,3,3,3,3,3,3,4,4,3,3,4,3,3,3,

%T 3,3,3,3,3,3,3,3,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

%N a(n) = common difference giving the highest score in the game "Sequences" for a sequence with largest element n and as many elements as possible.

%C The aim of the game is to find, in the given numbered tiles, arithmetic sequences of at least three elements. The score for a sequence is (max element)*(common difference ^ num elements). This sequence gives the common difference which produces the best score for a sequence with largest element n, and containing as many elements as possible.

%C This sequence converges to 3, which must have something to do with the fact that log(x)/x is maximized at x=e. Is n=45 the last time a(n) is not 3?

%H Christian Perfect, <a href="http://christianp.github.io/sequences">Sequences game</a>

%o (Python)

%o def score(n,step):

%o ...return n*(step**(n//step+(1 if n%step!=0 else 0)))

%o .

%o def best(n):

%o ...return max(range(1,(n-1)//2+1),key=lambda step:score(n,step))

%K nonn,easy

%O 3,3

%A _Christian Perfect_, Jun 09 2015