A262339 Exceptional primes for Ramanujan's tau function.

2, 3, 5, 7, 23, 691

Offset: 1

Author

Jonathan Sondow, Sep 18 2015

Keywords

• nonn
• fini
• full

Comments

For each exceptional prime p, Ramanujan's tau function tau(n) = A000594(n) satisfies a simple congruence modulo p.

The main entry for this subject is A000594.

Terms 23 and 691 also appear in A193855. - Jud McCranie, Nov 05 2020

Examples

			691 is an exceptional prime because tau(n) == sum of 11th power of divisors of n mod 691 (see A046694).
		

References

• H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.

Links

• H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
• Wikipedia, Ramanujan tau function

Crossrefs

Cf. A000594, A046694, A193855.