First Occurrences in OEIS A335901

OEIS A335901, %N (verbatim):

a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1.

%C (verbatim):

Least k such that a(k) = 2^n are 1, 2, 5, 21, 169, 2705, ... (Conjecture: This sequence is A117261).

The comparison sequence A117261 is given by A117261(0) = 1 and A117261(r+1) = 2^r * A117261(r) + 1. The phrase "least k such that a(k) = 2^r" is formalized as a hit together with minimality among positive indices.

The conjecture identifies the first occurrence of every power of two in a self-referential recurrence with a named OEIS sequence. A proof settles the conjecture and records the resulting open-problem resolution.

The dated surfaces recorded from issue #7468 and the probe are: OEIS history revision #4 introduces the conjecture, revision #5 rewords it, and revision #16 leaves it unchanged. A117261 revision #18 does not mention A335901. OpenAlex, Crossref, DataCite, arXiv, MathOverflow, and Math.StackExchange searches returned 0 exact matches for the paired claim. GitHub code search for both A-numbers returned 0 results. These are bounded search results, not an exhaustive literature claim.

The proof establishes the block invariant: a(T(r)) = 2^r; for r >= 1 and T(r) <= n < T(r+1), a(n) is either 2^(r-1) or 2^r; and if a(n) = 2^(r-1), then n < 2^(r-1) * T(r). A two-step induction on r places the recurrence quotient into one of the preceding two blocks. The hit occurs at the block start, and the block bound supplies minimality.

One natural r for which the least positive k with a(k) = 2^r differs from T(r) would falsify the theorem.

• Lean module: D5/S1/Recurrence/Invariants/SelfReferentialDoublingFirstOccurrence.lean.
• Main theorem: alkan_a335901; the theorem states the hit and minimality for every natural r.
• Definitions: a and T; the private block invariant is an internal proof component.
• Axioms: std3 (propext, Classical.choice, Quot.sound).
• Orchestrator recurrence to n = 3*10^6: first occurrences of 2^r for r = 0..6 are 1, 2, 5, 21, 169, 2705, 86561, and all observed values are powers of two.
• Probe to n = 6*10^6 reaches r <= 7, with the r = 7 first occurrence at 5539905; the b-file check reports 10000/10000.

theorem

None beyond the bounded literature search described above.