The arithmetic properties of the second order mock theta function \(\mathcal {B}(q)\) , introduced by McIntosh, defined by \(\begin{aligned} \mathcal {B}(q) := \sum _{n \ge 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum _{n \ge 0}b(n)q^n, \end{aligned}\) have been extensively studied. For instance, for all \(n\ge 0\) , Kaur and Rana [9] established congruences such as \(\begin{aligned} b(12n+10)&\equiv 0 \pmod {36}, \quad b(18n+16) \equiv 0 \pmod {72}, \end{aligned}\) Chen and Mao [6] proved that for all \(n\ge 0\) , \(\begin{aligned} b(4n+1)&\equiv 0 \pmod {2}, \quad b(4n+2) \equiv 0 \pmod {4}, \end{aligned}\) while Mao [10] also showed that for all \(n\ge 0\) , \(\begin{aligned} b(6n+2)&\equiv 0 \pmod {4}, \quad b(6n+4) \equiv 0 \pmod {9}. \end{aligned}\) In this paper, we find new congruences and infinite families of congruences modulo 4, 6, 36, 54, 72 for the function \(\mathcal {B}(q)\) . For example, let \(p \ge 5\) be a prime and \(1 \le \ell \le p - 1\) such that \(\left( \frac{12\ell + 9}{p} \right) _L = -1\) . Then for all \(n, k \ge 0\) , we have \(\begin{aligned} b\left( 6p^{2k+3}n + \frac{3p^{2k+2}(4\ell +3)-1}{2}\right) \equiv 0 \pmod {36}. \end{aligned}\) Our techniques involve elementary q-series and Maple.
