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 \( 10^n/p\) is prime, with value 0 if no such prime exists. We prove that \(a(n)>0\) for every \(n≥ 1\), 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 \(n≥ 2\), the prime quotient 2 is attained by a prime denominator. We also investigate the growth of the least admissible denominator. For \(L(K)=lcm\{p-1:p≤ K,\ p prime\}\), every positive multiple \(n\) of \(L(K)\) satisfies \(a(n)>K\) for \(K≥ 5\). Consequently, every fixed threshold is exceeded on a set of positive lower density and \(n→∞a(n)=∞\). An upper-bound sieve estimate for \(L(K)\) further yields\[n→∞a(n)/log n=∞.\]Using Jia's exceptional-set theorem for primes in short intervals, we additionally prove that for every fixed \(0<β<19/39\),\[a(n)=o\!(10(1-β)n),\]and hence\[n→∞{log₁₀a(n)}{n}≤20/39.\] The accompanying supplementary archive provides source code and full-range computational checks for the currently available OEIS b-file values \(n=1,…,1800\). These computations are logically separate from the proofs of the infinite and asymptotic results.