Even values of Ramanujan's tau-function
Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai

TL;DR
This paper investigates which even integers are not values of Ramanujan's tau-function, providing explicit examples and infinite families of such integers using recent advances in number theory.
Contribution
It offers the first explicit examples of even integers not attained by the tau-function and extends results to infinite families for primes less than 100.
Findings
Certain even integers are proven not to be tau-values.
Explicit non-tau even integers involving primes less than 100.
Infinite families of non-tau integers for powers of these primes.
Abstract
In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer is a value of . For odd , Murty, Murty, and Shorey proved that for sufficiently large . Several recent papers have identified explicit examples of odd which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes we find that Moreover, we obtain such results for infinitely many powers of each prime . As an example, for we prove that $$\tau(n)\not…
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