Exact p-Adic Arithmetic of Weighted Exponential Series and Divisor-Sum Euler Products

We study the arithmetic of two families of formal exponential power series. Weighted cycle expansions and a local Euler-product criterion yield exact p-adic valuations, reduced denominators, and a signed divisor-sum valuation, with specializations to OEIS A155200, A174491 and A262826. For a second family with logarithmic coefficients σ(2nᵈ)-σ(nᵈ), we give an explicit positive-integer formula for all Euler-product exponents and prove a sharp parameter-uniform binary congruence for every even d. It provides exact coefficient valuations for A195584 and A215603 and proves the original A224902 congruence conjecture, in fact modulo $8$. All statements hold for their full stated index ranges; independently checked finite examples are illustrations, not parts of the proofs. We distinguish the proven arithmetic conclusions from previously established integrality results and source identities. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.