A391359 - OEIS

%I #44 Jun 17 2026 17:38:13

%S 1,98,12,56292,361786618

%N a(n) is the period of the first differences of S_n(k) where the sequence S_n(k) is the trajectory of 1 under repeated application of the map: m -> m + n-th-smallest number that does not divide m, or -1 if no such orbit exists.

%C If an orbit with periodic first differences for S_n(k) exists:

%C   S_n(k) = S_n(k-1) + S_n(k-a(n)) - S_n(k-a(n)-1), for k > some offset.

%C   S_n(k+a(n)) = a(n) + c_n, for k > some offset. It is known that c_n is a term from A003418.

%C   The first few c_n for n=1..5 are: 2, 420, 60, 360360, 2677114440.

%C Other start values may reach eventually the same orbit in at least many if not all cases.

%e a(2) = 98, because A140485(n + 98) = A140485(n) + 420 for n >= 10, and A140485 is the trajectory of 1 under repeated application of the map: n -> n + second-smallest number that does not divide n.

%e a(3) = 12, because A140490(n + 12) = A140490(n) + 60 for n >= 13, and A140490 is the trajectory of 1 under repeated application of the map: n -> n + third-smallest number that does not divide n.

%Y Cf. A003418.

%Y Cf. A140485-A140494.

%K nonn,more

%O 1,2

%A _Thomas Scheuerle_, Jun 06 2026