The multiplicative \(d\)-dimensional partition of the positive integer \(n\) is a \(d\)-dimensional array of positive integers \(n_{i_1i_2\dots i_d}\) satisfying \(n_{j_1 j_2\dots j_d}\ge n_{k_1 k_2\dots k_d}\) if \(j_1\le k_1, j_2\le k_2,\dots j_d\le k_d\), and such that ther product equals \(n\). The Touchard polynomial \(T_n(x)\) of \(n\)th degree is defined by the exponential generating function \[ e^{x(e^t-1)}=\sum_{n=0}^\infty\frac{T_n(x)}{n!}t^n, \quad T_0(x):=1. \] Clearly, for all \(n\ge 0\), \(T_n(1)=\mathcal{B}_n\), where \(\mathcal{B}_n\) denotes the \(n\)th Bell number.
In this paper, the author deals with the number \(\mu_d(n)\) of \(d\)-dimensional partitions of the product \(\prod_{i=1}^n p_i\), where \(p_i\) denotes the \(i\)-th smallest prime number. It is known that \(\mu_1(n)=\mathcal{B}_n\). The author uses a combinatorial argument and generating functions to show that, for any integer \(n>0\), \(\mu_2(n)=2^n T_n(1/2)\). In addition, he conjectures that \(\mu_3(n)\le 3^n T_n(1/3)\).