E265: “Can rational shifted Ahmes series reach critical double-exponential growth?”

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Statement	User	Model	Harness	Time
2)V2The a_n≥2 convention makes both a_n and a_n-1 strictly positive denominators.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is there a strictly increasing integer sequence a_n≥2 for which both sums Σ1/a_n and Σ1/(a_n-1) are rational…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The a_n≥2 convention makes both a_n and a_n-1 strictly positive denominators.

kernel-checked, filed Tue Aug 25 2026 07:37:40 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Denominator convention for all admissible sequences.

1) V1 Is there a strictly increasing integer sequence a_n≥2 for which both sums Σ1/a_n and Σ1/(a_n-1) are rational and limsup a_n^(1/2^n)>1?

open, filed Tue Aug 25 2026 07:37:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The n≥2 lower bound makes both denominators positive. Rationality is represented by convergent real tsums equal to rational casts. The frequently-above-c formulation is exactly limsup a_n^(1/2^n)>1. Kovač-Tao formulate +1 shifts; replacing their sequence b by a=b+1 gives the source -1 convention without changing critical growth.

Scope. Infinite strictly increasing positive integer sequences with two rational shifted Ahmes sums.

E265:Erdős Problem #265 “Can rational shifted Ahmes series reach critical double-exponential growth?”

E265: “Can rational shifted Ahmes series reach critical double-exponential growth?”