Claims

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1965_03_01_erdos: Erdős's 1965 theorem that every sequence has $A_k>c\log k$ for infinitely many $k$, so $\limsup_k A_k=\infty$; the first answer to the first question, refereed in the Israel Journal of Mathematics and credited by the curator.

1967_01_01_clunie: Clunie's 1967 theorem that every sequence has $A_k\gg k^{1/2}$ for infinitely many $k$, so $\limsup_k A_k=\infty$; it answers the first question and is refereed in the Journal of the London Mathematical Society.

1969_06_01_liu: Liu's 1969 theorem that a sequence taking finitely many distinct values has $\limsup_kA_k/k^{1-\delta}>1/5$ for every $\delta>0$; the first question for such sequences, refereed in Proc. Amer. Math. Soc.

2025_08_30_tao: Tao's 2025 forum proof that $\limsup_k A_k=\infty$ for every sequence, with a Lean 4 formalization in his analysis repository, not built here; a reproof of the first question's answer, credited by the site's curator.

2026_04_08_alexeev_putterman_sawhney_sellke_valiant: Alexeev, Putterman, Sawhney, Sellke and Valiant's 2026 theorem, due to an internal OpenAI model, that some sequence has $A_k\ll\sqrt{k\log 2k}=o(k)$ for all $k$, answering the second question; accepted by the site's curator.

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