Growth of the Mian-Chowla (greedy Sidon) sequence: compute terms and measure the exponent (Erdos #340)

The Mian-Chowla sequence (OEIS A005282) is the greedy Sidon sequence: a_1=1, and for n>1, a_n is the smallest integer > a_(n-1) such that {a_1,...,a_n} remains a Sidon set (no nontrivial solution to a_i + a_j = a_k + a_l). It begins 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, ... Erdos #340 asks for the order of growth of A(N) = |{a_i} intersect {1,...,N}|; in particular whether A(N) >>_eps N^(1/2 - eps) for all eps>0. The greedy sequence is far sparser than optimal Sidon sets (which reach ~N^(1/2)), and its true growth exponent is unknown. Computational sub-questions: (a) compute the sequence to large N with a reproducible generator and extend the published terms; (b) tabulate A(N), A(N)/N^(1/2), and A(N)/N^(1/3) over a wide range and report the empirical exponent log A(N)/log N; (c) obtain any rigorous upper or lower bound on the growth.

Acceptance. FULLY RESOLVES: a proof of the exact order of growth of A(N). COMPUTATIONAL PROGRESS (in scope): (a) extension of A005282 to substantially more terms than currently published, with the generating program so the sequence can be re-derived; (b) a tabulated/plotted trend of A(N) and A(N)/N^(1/2) and A(N)/N^(1/3) over a wide range of N, with a stated empirical exponent and confidence interval; (c) any rigorous bound with a certificate. All submissions must include the deterministic generator and the computed terms so the result is independently reproducible.