Geodesic
Derived sequences
Journal of integer sequences, Tome 6 (2003) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We define a multiplicative arithmetic function $D$ by assigning $D(p^a)=$ap^a-1, when $p$ is a prime and a is a positive integer, and, for $n >= 1$, we set D^$0(n)=n$ and $D^k(n)=$D(D^k-1$(n))$ when $k>= 1$. We term ${D^k(n)}$_k >= 0 the derived sequence of $n$. We show that all derived sequences of $n 1.5 * 10$^10 are bounded, and that the density of those $n$ in N with bounded derived sequences exceeds 0.996, but we conjecture nonetheless the existence of unbounded sequences. Known bounded derived sequences end (effectively) in cycles of lengths only 1 to 6, and 8, yet the existence of cycles of arbitrary length is conjectured. We prove the existence of derived sequences of arbitrarily many terms without a cycle.
Classification : 11Y55, 11A25, 11B83
Keywords: arithmetic functions, density, unbounded sequences, cycles 
@article{JIS_2003__6_1_a2,
     author = {Cohen, C.L. and Iannucci, D.E.},
     title = {Derived sequences},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {6},
     number = {1},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2003__6_1_a2/}
}
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Cohen, C.L.; Iannucci, D.E. Derived sequences. Journal of integer sequences, Tome 6 (2003) no. 1. http://geodesic.mathdoc.fr/item/JIS_2003__6_1_a2/