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For given positive integers $a_1,a_2,\dots,a_k$ with$\gcd(a_1,a_2,\dots,a_k)=1$, the denumerant $d(n)=d(n;a_1,a_2,\dots,a_k)$ isthe number of nonnegative solutions $(x_1,x_2,\dots,x_k)$ of the linearequation $a_1 x_1+a_2 x_2+\dots+a_k x_k=n$ for a positive integer $n$. For agiven nonnegative integer $p$, let $S_p=S_p(a_1,a_2,\dots,a_k)$ be the set ofall nonnegative integers $n$'s such that $d(n)>p$. In this paper, we areinterested in the $p$-Frobenius number, which is the maximum of the set of gaps$\mathbb N_0\backslash S_p$. Here $\mathbb N_0$ denotes the set of nonnegativeintegers. When $p=0$, $S=S_0$ is the original numerical semigroup, and the$0$-Frobenius number is the original Frobenius number. The explicit formula fortwo variables is known not only for $p=0$ but also for $p>0$, but when thereare three or more variables, it is difficult even in the special case of $p=0$.For $p>0$, it is not only more difficult, but no explicit formula had beenfound. In this paper, explicit formulas of the $p$-Frobenius number and relatedvalues are given for the triple of arithmetic progressions. The main tool is todetermine the elements of the $p$-Ap\'ery set.

The $p$-numerical semigroup of the triple of arithmetic progressions