How many maps of Z/NZ preserve multiplication? A counting formula for OEIS sequence A117986

Sequence A117986 of the On-Line Encyclopedia of Integer Sequences (OEIS) gives the number N(n) of maps f: Z/nZ → Z/nZ with f(xy) = f(x)f(y) for all x and y. The entry lists 56 terms and no proved formula; its only formula line, added in 2019, says that "apparently" N(p) = p + 1 for every prime p. This paper confirms that remark with a short proof and gives a formula for N(n) for every n. By the Chinese remainder theorem the count splits into local numbers T(pa, qb), the numbers of multiplicative maps g: Z/paZ → Z/qbZ with g(1) = 1, and N(n) = ∏i (1 + ∏j T(qj, qi)), where q1, ..., qr are the prime powers in the factorisation of n. Each local number is a finite sum of counts of solutions of explicit congruences; the prime 2 is treated separately, because the units modulo 2k do not form a cyclic group for k ≥ 3. This gives N(pk) = p2k−2(p − 1) + 2 for odd p, N(2k) = 3·22k−2 + 2 for k ≥ 3, and a closed formula for square-free n. The formula agrees with all 56 published terms and with exhaustive search for every n ≤ 72, and it gives 16 new terms. The maps between prime-power monoids had already been described in a 2026 doctoral thesis by J. Atalaye (Stellenbosch University), whose explicit forms imply the local counts in many cases; for Z/8Z, however, one of its corollaries implies 65 maps with g(1) = 1, while a direct enumeration gives 49. The contribution of this paper is an element-by-element count of these maps for every pair of prime powers, checked against direct counts, the resulting formula for N(n), and its connection to A117986.