Integer complexity satisfies ||n|| ≤ C_avg log n + o(log n) implying lim sup ||n||/log n ≤ C_avg ≈ 3.236, plus the first nontrivial lower bound ||n|| ≥ 3.06 log_3 n for almost all n.
Steinerberger,A short note on integer complexity, Contrib
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Upper and lower estimates for integer complexity
Integer complexity satisfies ||n|| ≤ C_avg log n + o(log n) implying lim sup ||n||/log n ≤ C_avg ≈ 3.236, plus the first nontrivial lower bound ||n|| ≥ 3.06 log_3 n for almost all n.