The Sisyphus Sequence (A350877)

(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)

Sisyphus, in Greek mythology, was cursed upon death to roll a large boulder up a hill only for it to roll down once it neared the top, repeating for eternity. 

The integer sequence, OEIS:  (https://oeis.org/A350877) starts 1, 3, 6, 3, 8, 4, 2, 1, 8 ,4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1... . The formal definition for the sequence is: start the sequence S with a(1) = 1 and extend S with a(n)/2 when a(n) is even, otherwise with a(n) + the smallest prime not yet added. The sequence was named Sisyphus because the sequence climbs only to drop back to 1.

Additional notes on the sequence:  Michael De Vlieger's OEIS sequence analysis: A350877.

A proposed variant of this sequence is to begin the sequence with n, integers 1 and greater, and define a(n) as the number of steps required to reach 1. This new sequence is: 

0, 1, 5, 2, 979, 6, 8, 3, 13, 980, 6, 7, 209, 9, 980, 4, 8, 14, 9, 981, 10, 7, 14, 8, 980, 210, 7, 10, 20, 981, 210, 5, 14, 9, 981, 15, 13, 10, 9, 982, 210, 11, 10, 8, 806, 15, 11, 9, 9, 981, 15, 211, 980, 8, 981, 11, 11, 21, 8, 982, 16, 211, 21, 6, 981, 15, 211...

a(1) = 0 because no further steps are required to reach 1.

a(2) = 1 because in one step, 2/2 =1, 1 is reached.

a(3) = 5 the steps proceed: 5, 8, 4, 2, 1 (5 steps to reach 1).

An interesting feature of this new sequence is how terms tend to cluster around certain values - from just the first 67 terms above, see terms clustered around 11, 210, 980.

This clustering feature is shown below in the scatterplot of the first 4000 terms:

Michael De Vlieger produced a log-log scatterplot of the first 65536 terms, which shows how this clustering continues:

Sequence published: https://jamesmacmath.blogspot.com/2025/09/a388038-contribution-to-oeis-is-number.html.