Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

Fix a prime $p\ge 5$ and define $g(2n,p)=\mathrm{\#}\{(h,k)\in {\mathbb{Z}}_{>0}^2:h+k=2n,h\le k,\gcd (h,6p)=\gcd (k,6p)=1\}$. We derive explicit closed-form expressions for $g(2n,p)$ in terms of the canonical remainder operator ${\delta }_k(x)=x-k\lfloor x\mathrm{/}k\rfloor$, elementary step functions, and the minimal solutions of the congruences $6x\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}p)$ and $6x\equiv -5(\mathrm{m}\mathrm{o}\mathrm{d}p)$. A key ingredient is an explicit formula for the minimal solution of ${\delta }_k(a_{0}x)=b_0$ obtained via the Euclidean algorithm, which determines the excluded residue classes directly. The resulting formulas show that $g(2n,p)$ is piecewise affine along arithmetic progressions of $n$, governed by residue classes modulo $3$ and $p$. For fixed $p$, after precomputing two residue parameters in $O(\log p)$ time, each evaluation of $g(2n,p)$ requires only $O(1)$ operations, compared to $O(n)$ for direct enumeration. The formulas are validated computationally for all $2n\le {10}^5$ and primes $p\in \{5,7,11,13,17,19,23\}$, with perfect agreement with brute-force enumeration.