Congruences for Andrews spt-function modulo powers of \(5, 7\) and \(13\) (Q2841373)

Let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\) and let \(\delta_{\ell,k}\) denote the least non-negative residue of \(24^{-1}\) modulo \(\ell^k\). The author shows that NEWLINE\[NEWLINE \begin{aligned} \text{spt}(5^a n + \delta_{5,a}) &\equiv 0 \pmod{5^{\lfloor \frac{a+1}{2} \rfloor}}, \\ \text{spt}(7^b n + \delta_{7,b}) &\equiv 0 \pmod{7^{\lfloor \frac{b+1}{2} \rfloor}}, \\ \text{spt}(13^c n + \delta_{13,c}) &\equiv 0 \pmod{13^{\lfloor \frac{c+1}{2} \rfloor}}. \end{aligned} NEWLINE\]NEWLINE These are reminiscent of the partition congruences NEWLINE\[NEWLINE \begin{aligned} p(5^a n + \delta_{5,a}) &\equiv 0 \pmod{5^a}, \\ p(7^b n + \delta_{7,b}) &\equiv 0 \pmod{7^{\lfloor \frac{b+2}{2} \rfloor}}, \\ p(11^c n + \delta_{11,c}) &\equiv 0 \pmod{11^c}. \end{aligned} NEWLINE\]NEWLINE Indeed, the proof of the congruences for \(\text{spt}(n)\) ultimately uses the method of modular equations which \textit{G. N. Watson} [J. Reine Angew. Math. 179, 97--128 (1938; Zbl 0019.15302)] and \textit{A.~O.~L. Atkin} [Glasg. Math. J. 8, 14--32 (1967; Zbl 0163.04302), Can. J. Math. 20, 67--78 (1968; Zbl 0164.35101)] used to prove the congruences for \(p(n)\), but first the author skilfully applies a \(U\)-type operator to pass from a (quasi-)mock modular generating function to a modular one.