Abstract. Call a positive integer prime-prefix-free if none of its proper binary prefixes βπ/2πβ, π β₯ 1, is an odd prime (OEIS A287117). Assuming the Riemann Hypothesis, we prove that the sum of 1/π over the prime-prefix-free integers converges. More precisely, the proportion of [2π, 2π+1) they occupy is π πΏ (πβ1/log 2+ πΏ) for every πΏ > 0. The heuristic exponent is 1/log 2 β 1.4427, and ...