For what values of $t,\alpha \in (0,\infty )$ is the sequence $\lfloor t{\alpha }^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t{\alpha }^n\rfloor$)?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let [ a_n=\big\lfloor t,\alpha^n\big\rfloor\qquad (n=1,2,3,\dots) ] and let (\Sigma) be the set of all **finite sums of distinct terms** (a_{n_1}+\cdots+a_{n_k}) (distinct indices). The sequence is **complete** if every sufficiently large integer lies in (\Sigma).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.