{"schema":"vela.problem-packet.v0.1","problem":647,"statement":"Let $\\tau(n)$ count the number of divisors of $n$. Is there some $n&#62;24$ such that\\[\\max_{m&#60;n}(m+\\tau(m))\\leq n+2?\\]","status":"open","seam":"sealed","closureRoutes":[{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"},{"type":"new_channel_reduction","verifierKind":"lean","note":"the prime/forced-divisor channel is PROVABLY exhausted (banked obstruction_map + Erdos647Obstruction.lean) — only a different channel counts"}],"obligations":[{"findingId":"vf_ac4c84ea5e608bb7","banked":"the prime / forced-divisor channel is provably exhausted (banked obstruction_map + Erdos647Obstruction.lean)","open":"a different channel is required; the prime channel cannot close it (stop grinding it).","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_372e1dede866a97c","kind":"dead_end","claim":"Erdos #647 (VERIFIABLE): no n>24 with max_{m<n}(m+tau(m)) <= n+2 has been found. A deep verifier-gated search (Codex; prime-7 candidate classes + P8/P9 smooth reductions + residue-guided sieving) covered n up to ~3e11 with no witness. Independently spot-checked to 2e6: exactly n in {2,3,4,5,6,8,10,12,24} satisfy the property, with 24 the largest. This is a null result extending the searched range, not a proof of nonexistence.","grade":"honest_null","gateStatus":"verified","superseded":false},{"id":"att_4786e4516a93f968","kind":"dead_end","claim":"Erdős #647 (max_{m<n}(m+τ(m)) ≤ n+2 for some n>24) — OBSTRUCTION MAP, not a settlement. UNCONDITIONAL core (two independent recomputes + Lean): (U1) 2·τ(k) ≤ k+2 for all k≥1, with equality only at k∈{2,4,6}; so a prime-channel candidate with n−k=k·p forces τ(n−k)=2·τ(k) to graze but never exceed the k+2 ceiling, yielding no witness via that channel. (U2) No finite forced-divisor covering {n≡kᵢ mod Dᵢ, τ(Dᵢ)>kᵢ+2} covers all large n: τ(Dᵢ)>kᵢ+2 ⟹ Dᵢ>kᵢ ⟹ kᵢ≢0 mod Dᵢ, so multiples of lcm(Dᵢ) escape every class — the prime/forced-divisor channel cannot settle #647. CONDITIONAL (NOT asserted): whether any n>24 attains the property reduces to Schinzel–Dickson / prime k-tuple statements.","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos647Obstruction.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos647Obstruction.lean"}],"oeis":[{"id":"A062249","name":"a(n) = n + d(n), where d(n) = number of divisors of n, cf. A000005.","terms":"2,4,5,7,7,10,9,12,12,14,13,18,15,18,19,21,19,24,21,26,25,26,25,32,28,30,31,34,31,38,33,38,37,38,39,45,39,42,43,48,43,50,","url":"https://oeis.org/A062249"},{"id":"A087280","name":"Solutions n of max(m+d(m))=n+2 for m<n; d(m) is the number of divisors of m.","terms":"5,8,10,12,24","url":"https://oeis.org/A087280"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}