A391359 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.

1, 98, 12, 56292, 361786618

Offset

1,2

Comments

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

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

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

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

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

Links

Table of n, a(n) for n=1..5.

Example

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.
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.

Crossrefs

Cf. A003418.

Cf. A140485-A140494.

Sequence in context: A122128 A114430 A258765 * A033418 A120310 A129887

Adjacent sequences: A391356 A391357 A391358 * A391360 A391361 A391362

Keyword

nonn,more

Author

Thomas Scheuerle, Jun 06 2026

Status

approved