Schulte’s divisibility–primality conjecture for A006472

Establish that for every integer n > 1, the divisibility n | (2A_{n-1} + 4) holds if and only if n is prime, where A_k = k!(k−1)! / 2^{k−1}.

Background

The paper studies the integer sequence A_n defined as the product of the first n−1 triangular numbers, which can be written as A_n = (n−1)!·n! / 2^{n−1} and is listed in OEIS as A006472. On the OEIS page for this sequence, Werner Schulte proposed a conjecture connecting the divisibility of a specific expression involving A_{n−1} to the primality of n.

The authors highlight this conjecture and present an elementary proof within the paper, drawing on classical results such as Wilson’s theorem and Fermat’s Little Theorem. The conjecture asserts a precise equivalence: n divides 2A_{n−1} + 4 if and only if n is prime.