{"schema":"vela.problem-packet.v0.1","problem":409,"statement":"How many iterations of $n\\mapsto \\phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_913e8bc70f5f0fae","kind":"partial_proof","claim":"PROVED (kernel-verified, INDEPENDENTLY re-compiled) the termination sorry of Erdős #409: erdos_409.variants.termination — for every n>0 the iteration n->phi(n)+1 reaches a prime. (Closes one sorry in formal-conjectures; the hard parts of #409 remain open.)","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_cec63fe30c79261d","kind":"partial_proof","claim":"Proved (machine-checked in Lean/Mathlib, clean compile) the termination part of Erdős 409: for n>0 the φ-iteration n↦φ(n)+1 always reaches a prime, via strict descent on composites (φ(n)≤n-2 since φ(n","grade":"partial_proof","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A039651","name":"Number of iterations of f(x) = phi(x)+1 on n required to reach a prime.","terms":"1,0,0,1,0,1,0,1,1,1,0,1,0,1,2,2,0,1,0,2,1,1,0,2,2,1,1,1,0,2,0,1,2,1,3,1,0,1,3,1,0,1,0,2,3,1,0,1,1,2,3,3,0,1,1,3,1,1,0,1,","url":"https://oeis.org/A039651"},{"id":"A229487","name":"Conjectured greatest number that converges to prime(n) under the iteration x -> phi(x) + 1, where phi is Euler's totient function.","terms":"2,6,12,30,22,138,60,54,46,58,62,174,498,510,94,106,118,4314,134,142,1038,158,166,276,420,250,206,214,750,1758,254,262,27","url":"https://oeis.org/A229487"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}