TY  - JOUR
AB  - We define a multiplicative arithmetic function D by assigning D(p a) = apa-1, when p is a prime and a is a positive integer, and, for n ? 1, we set D0(n) = n and Dk(n) = D(D k-1(n)) when k ? 1. We term {Dk(n)}k=0? the derived sequence of n. We show that all derived sequences of n < 1.5 · 1010 are bounded, and that the density of those 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.
AU  - Cohen, GL
AU  - Iannucci, DE
DA  - 2003/12/01
JO  - Journal of Integer Sequences
PY  - 2003/12/01
TI  - Derived sequences
VL  - 6
Y1  - 2003/12/01
Y2  - 2026/05/21
ER  -