Minimal admissible Hilbert cubes

For positive integers a_1 < ... < a_n let Σ(a_1, ..., a_n) be the set of their 2^n subset sums. We call the cube admissible if the subset sums are pairwise distinct and, for every prime p, they miss at least one residue class modulo p. Let a(n) be the least possible value of a_1 + ... + a_n. This is the smallest diameter of a Hilbert cube that is not ruled out from consisting entirely of primes by congruence obstructions. We determine a(1), ..., a(7) = 2, 8, 32, 82, 302, 1238, 7792, each attained by a unique set of generators, and show a(8) ≤ 116306 by an explicit example. A search by one implementation gives a(8) ≥ 20002. We prove some elementary structural facts: a(n) is at least the minimal width of an admissible 2^n-tuple, and a residue class that is the only one left free forces every further generator to be divisible by that prime. The second fact explains why the minimizers are not nested: no extension of the n = 7 minimizer has diameter below 2 707 762. The sequence is OEIS A398220. Project website: https://stokrotki.eu