Nonvanishing and Growth of Least Prime Denominators for Decimal Powers

This preprint studies OEIS A090519, where \(a(n)\) is the least prime \(p\) such that \(\lfloor 10^n/p\rfloor\) is prime, with value 0 if no such prime exists. We prove that \(a(n)>0\) for every \(n\ge 1\), resolving the nonvanishing conjecture recorded for the sequence. An elementary proof uses Bertrand's theorem, while Nagura's theorem gives the stronger result that, for every \(n\ge 2\), the prime quotient 2 is attained by a prime denominator. We also investigate the growth of the least admissible denominator. For \(L(K)=\operatorname{lcm}\{p-1:p\le K,\ p\text{ prime}\}\), every positive multiple \(n\) of \(L(K)\) satisfies \(a(n)>K\) for \(K\ge 5\). Consequently, every fixed threshold is exceeded on a set of positive lower density and \(\limsup_{n\to\infty}a(n)=\infty\). An upper-bound sieve estimate for \(L(K)\) further yields\[\limsup_{n\to\infty}\frac{a(n)}{\log n}=\infty.\]Using Jia's exceptional-set theorem for primes in short intervals, we additionally prove that for every fixed \(0<\beta<19/39\),\[a(n)=o\!\left(10^{(1-\beta)n}\right),\]and hence\[\limsup_{n\to\infty}\frac{\log_{10}a(n)}{n}\le\frac{20}{39}.\] The accompanying supplementary archive provides source code and full-range computational checks for the currently available OEIS b-file values \(n=1,\ldots,1800\). These computations are logically separate from the proofs of the infinite and asymptotic results.