A note on my 2012 paper "Patterns related to the Smarandache circular sequence primality problem"

This short note restates, in a compact and rigorous way, one of the main results first presented in Ripà (2012), Patterns related to the Smarandache circular sequence primality problem, published in Notes on Number Theory and Discrete Mathematics. Relying on the circular permutations of the concatenated sequence S(r) = 123...r (see OEIS A007908 for the base concatenation and A001292 for its circular permutations), that work examined the distribution of prime numbers within those arrangements, with particular attention to the positions of terms divisible by fixed primes. By constructing an ad hoc modular sieve that filtered out all multiples of smaller primes, the paper revealed, through explicit computation and graphical representation, the presence of perfectly periodic modular "tiles" describing the divisibility patterns arising from such rotations. Here, the same phenomenon is reformulated in modern terms, showing how it follows from the modular properties of decimal concatenation, repunits, and multiplicative orders.