Three OEIS Entries, Four Conjectures: Elementary Proofs of Prime-Related Number-Theoretic Statements

This paper presents elementary proofs of four conjectural assertions recorded in three entries of the Online Encyclopedia of Integer Sequences (OEIS): A280258, A228558, and A390815. For A280258, we establish an exact parity characterization for a divisor sum involving products of integers coprime to their moduli, proving that the sum is even if and only if its argument is twice a perfect square, \( n=2m^2 \). For A228558, we prove that whenever \( (17^k+4^k)/21 \) is prime, the exponent \( k \) must itself be prime. The proof relies on elementary factorization identities for sums of odd powers. For A390815, we establish both conjectured properties of the reduced denominator of \( n!/σ(n) \), where \( σ(n) \) denotes the sum-of-divisors function. We prove that every nontrivial denominator is prime and that, for \( n≥4 \), such denominators occur only when \( n=p²ᵏ \) for some prime \( p \) and positive integer \( k \). Moreover, we establish the stronger identity \( den(n!/σ(n))=σ(n) \) when \( σ(n) \) is prime, and \( den(n!/σ(n))=1 \) otherwise, for all \( n≥4 \). All proofs are elementary and apply to the full stated ranges, going beyond finite computational verification. The results demonstrate how classical divisibility arguments, multiplicative functions, and algebraic factorizations can resolve conjectural statements concerning prime numbers and integer sequences. The manuscript provides complete mathematical arguments and references to the corresponding OEIS entries. No claim of mathematical priority or previously unpublished discovery is made.