On the Infinitude of a Prime-Factor Recurrence

Abstract

We study the prime-factor recurrence recorded as OEIS A242462, starting from 10 and using the least odd prime divisor at each step. Following the squarefreeness observation and conditional criteria of J. Lowell, we show that no term is a power of two; hence the recurrence continues indefinitely and is strictly increasing. The proof uses persistent divisors of the sixth term minus one and a congruence modulo \(13^2\). Three examples illustrate the recurrence and the finite arithmetic; an appendix records exact terms through index 20 with independently checkable factor data.