On some values which do not belong to the image of Ramanujan’s tau-function

Lehmer conjectured that Ramanujan’s tau-function never vanishes. As a variation of this conjecture, it is proved that

$$\begin{aligned} \tau (n)\ne \pm \ell , \pm 2\ell , \pm 2\ell ^2, \end{aligned}$$

where \(\ell <100\) is an odd prime, by Balakrishnan, Ono, Craig, Tsai, and many people. We prove that

$$\begin{aligned} \tau (n)\ne \pm \ell , \pm 2\ell , \pm 4\ell , \pm 8\ell \end{aligned}$$

for \(\ell \in L\), where L is an explicit finite subset of odd primes less than 1000.