{"schema":"vela.problem-packet.v0.1","problem":417,"statement":"Let\\[V'(x)=\\#\\{\\phi(m) : 1\\leq m\\leq x\\}\\]and\\[V(x)=\\#\\{\\phi(m) \\leq x : 1\\leq m\\}.\\]Does $\\lim V(x)/V'(x)$ exist? Is it $&#62;1$?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_0167190a509a4a4f","kind":"partial_proof","claim":"Erdős 417 (does lim V(x)/V'(x) exist and exceed 1) remains OPEN, but I produced a clean reduction: proved unconditionally V'(x)<=V(x) (so ratio>=1) via φ(m)<m, the exact identity V−V'=#{totients n<=x ","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_erdos417_hidden_totient_decomposition","kind":"vela.erdos_problem_attempt_record_draft.v1","claim":"Erdos #417: lean/Vela/Erdos417.lean formalizes the finite-set comparison V'(x)<=V(x) and proves the exact decomposition V(x)=V'(x)+hiddenCount(x), where hiddenCount counts totient values v<=x whose preimages all lie above x. This isolates the official residual as an asymptotic question about hidden totients and does not prove the limit exists or exceeds 1.","grade":"partial_proof","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos417.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos417.lean"}],"oeis":[{"id":"A061070","name":"Number of distinct values in the list of values of the Euler totient function {phi(j) : j=1..n}.","terms":"1,1,2,2,3,3,4,4,4,4,5,5,6,6,7,7,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,14,14,15,15,15,15,16,16,17,17,17,17,18,1","url":"https://oeis.org/A061070"},{"id":"A264810","name":"Number of numbers k <= n such that phi(m) = k for some m.","terms":"1,2,2,3,3,4,4,5,5,6,6,7,7,7,7,8,8,9,9,10,10,11,11,12,12,12,12,13,13,14,14,15,15,15,15,16,16,16,16,17,17,18,18,19,19,20,2","url":"https://oeis.org/A264810"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}