Least Integer Witnesses for Prime Floor Quotients of Decimal Powers

For each integer n ≥ 1, let a(n) be the least positive integer k such that ⌊10^n/k⌋ is prime; this is the sequence OEIS A090517. We first prove directly that the defining set is nonempty for every n, so the least witness is always well-defined. We then give an exact characterization of the minimum. If Q_n is the largest prime q for which the quotient fiber (10^n/(q+1), 10^n/q] contains an integer, then a(n) = ⌊10^n/(Q_n+1)⌋ + 1. Finally, we prove the uniform elementary bound a(n) ≥ 7 for every n ≥ 2 by excluding all smaller denominators. No theorem on the distribution of primes is required. Separately, the 65 values currently supplied by OEIS are recomputed directly from the least-witness definition as a finite consistency check.