lean_workbook_plus_69984

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$ , $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$ . How many times $111$ appears on this sequence?

Why this node was retired

The posted statement is

theorem lean_workbook_plus_69984 (A : ℕ → ℕ) (hA: A 0 = 1 ∧ ∀ n:ℕ, A (2 * n) = A n + 1 ∧ A (2 * n + 1) = 10 * A n) : ∃ n:ℕ, A n = 111 ∧ ∀ k:ℕ, k < n → A k ≠ 111   :=  by sorry

The index-zero recurrence says A(0)=A(0)+1. The intended count is not determined by this audit.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Set A(1)=1 and impose both recurrences for n≥1 only; ask for the cardinality of{n≥1|A(n)=111}, as the source requests, rather than just a least occurrence.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.