  We count the numbers of associated primes of powers of ideals as defined by
Bandari, Hibi, and Herzog in 2014. We generalize those ideals to monomial
ideals $\operatorname{BHH}(m,r,s)$ for $r \ge 2$, $m$, $s \ge 1$; we establish
partially the associated primes of powers of these ideals, and we establish
completely the depth function of quotients by powers of these ideals: the depth
function is periodic of period $r$ repeated $m$ times on the initial interval
before settling to a constant value. The number of needed variables for these
depth functions are lower than those from general constructions by H\`{a},
Nguyen, Trung, and Trung (2021).
