We show that the number of primes with leading 1-run length exactly g in the L-bit range [2^ (L-1), 2L) is given by count (g, L) = integral from a to b of dt/ln (t), where [a, b) = [2L - 2^ (L-g), 2L - 2^ (L-g-1) ). A closed-form approximation requiring only one division and one logarithm is also derived: count (g, L) = 2^ (L-g-1) / ln (2L - 3 * 2^ (L-g-2) ). Verified by exhaustive sieve for L = 10 to 32 (primes up to 4. 3 billion), accuracy reaches 99. 99%. For higher precision, Riemann's R function can be applied to each sub-interval. The practical application of this result is, at this point, unknown to the author. Includes LaTeX source and a self-contained Python verification script.
Yukihiro Honda (Sun,) studied this question.