Tau Function Prime

 is prime for , 458329, 942841, 966289, 1510441, ... (OEIS A135430). These values are also known as Lehmer-Ramanujan numbers or LR numbers since the first of them was found by Lehmer (1965). The corresponding primes have explicit values given by , , ... (OEIS A265913).

It is known that if  is prime, then  must be an odd square.

Large values of  for which  is a (probable) prime are summarized in the table below (Lifchitz and Lifchitz).

	decimal digits	discoverer
	180524	N. Lygeros and O. Rozier (May 2015)
	258571	N. Lygeros and O. Rozier (May 2015)
	282048	N. Lygeros and O. Rozier (May 2015)
	498503	N. Lygeros and O. Rozier (May 2015)
	555339	N. Lygeros and O. Rozier (Sep. 2015)

REFERENCES:

Lehmer, D. H. "The Primality of Ramanujan's Tau-Function." Amer. Math. Monthly 72, 15-18, 1965.

Lifchitz, H. and Lifchitz, R. "PRP Records: Probable Primes Top 10000." https://www.primenumbers.net/prptop/searchform.php?form=Tau%28%3F%29&action=Search.

Sloane, N. J. A. Sequences A135430 and A265913 in "The On-Line Encyclopedia of Integer Sequences."