(Image: Michael De Vlieger)
I recently proposed a new sequence for the On-Line Encyclopedia of Integer Sequences (OEIS).
A Sisyphus sequence: a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive integer having the same greatest prime factor as the sum of the previous terms.
Sisyphus is a figure in Greek mythology, known as the cunning King of Corinth who was punished by Zeus to eternally roll a massive boulder up a hill in Tartarus, only for it to roll back down each time. Numerical sequences are sometimes called Sisyphus sequences when their terms climb and climb, but then drop repeatedly.
The image above shows this process for this sequence for the first 10^6 terms. Below is a graph for how the sequence begins (first 300 terms):
One interesting feature of the sequence is the low points all appear to be prime. They begin: 7, 13, 17, 19, 23, 29, 59, 107, 137, 173, 257, 293, 467, 503...
This trend continues for at least the first 200,000 terms of the sequence.
Another feature of the sequence is that beginning with the third term, the common greatest prime factor repeats in runs of 3 or more. These factors begin: 3, 3, 3, 7, 7, 7, 7, 13, 13, 13, 13... and this trend continues for at least the first 200,000 terms of the sequence.
The proposed sequence is currently in draft form in the OEIS: https://oeis.org/draft/A392975.
(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)
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