A New OEIS Candidate Extending A082772: A Full First-Return Divisor Sequence

Based on previous Zenodo Preprint " Prime-Level First-Return Densities: A DiagonalDivisor Transport Framework" ,this work introduces a new integer sequence proposed as a candidate for inclusion in the On-Line Encyclopedia of Integer Sequences. It arises from a natural extension of OEIS A082772, which appears here as the first level of a broader arithmetic structure rather than as an isolated divisibility condition. The new sequence records the first displacement at which a nontrivial divisor relation reappears when an integer and one of its divisors are shifted simultaneously. This simple construction generates a complete hierarchy of first-return levels and reveals a remarkably structured arithmetic landscape. The sequence is supported by exact theoretical results. Its exceptional values are completely characterized, its finite values satisfy a sharp global bound, odd prime squares produce extremal values, and every fixed divisor-return mechanism generates an explicit arithmetic progression. The associated return levels possess natural densities forming a normalized distribution, while prime-indexed levels exhibit a distinctive asymptotic behavior. The connection with OEIS A082772 is particularly significant: A082772 is recovered exactly as the first-return level of the new sequence. The present construction therefore does not merely resemble an existing OEIS entry; it places that entry inside a wider and mathematically coherent first-return framework. A computation of one million terms has been carried out using the exact progression structure and independently checked against the direct divisor definition on an initial range. No corresponding OEIS entry for the complete first-return function was identified in searches performed in August 2026. The resulting sequence therefore appears to provide a natural and nontrivial extension of a previously catalogued OEIS phenomenon, combining a simple definition with rigorous arithmetic structure, large-scale computational evidence, and several new questions concerning its range, records, level densities, and prime-related behavior. Sylvain GeffroyIndependent ResearcherCaen, France