# A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded because $(2,6,18)=(2,2 {\cdot} 3, 2 {\cdot} 3^2)$ forms a geometric progression. He showed that the asymptotic density of that greedy sequence exceeds $0.71$.

I wondered about sequences that avoid 3-term power progressions, $(a, a c, a c^k)$, where $c\ge 2$ and $k\ge 2$ are natural numbers. Here I start again with $(1,2)$ and continue to add the first number that avoids all 3-term power progressions: $$\{1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, \ldots \}\;.$$ For example, $8$ is excluded because $(1,2,8)=(1,1{\cdot} 2, 1 {\cdot} 2^3)$ is a power sequence. Much later, $945$ is excluded because $(35,105,945)=(35, 35{\cdot} 3, 35 {\cdot} 3^3)$.

Q. What is the asymptotic density of the above greedy sequence that avoids all 3-term power progressions?

The density appears to be about $0.63$, up to $644$ terms ending in $1021$. Note that the square-free integers have density $6/\pi^2 \approx 0.61$. My sequence is square-free, cube-free, etc., because $(1, c, c^k)$ is a power progression.

1R. Rankin. "Sets of integers containing not more than a given number of terms in arithmetical progression." Proc. Roy. Soc. Edinburgh Sect. A. 65 (1961). Cited by Nathan McNew in poster, "Avoiding Geometric Progressions in the Integers." (PDF download.)

• What is the question? – Masacroso Jun 30 '15 at 13:36
• @Masacroso: What is the asymptotic density. My simulations are hardly definitive. Or bounds on the density. – Joseph O'Rourke Jun 30 '15 at 13:40