A193855 -id:A193855 - cp's OEIS frontend

A295645 Primes p such that tau(p) +- 1 is congruent to 0 (mod p), where tau is the Ramanujan tau function (A000594).

11, 23, 691, 5807

Offset: 1

Seiichi Manyama, Nov 25 2017

Nik Lygeros and Olivier Rozier found a new solution to the equation tau(p) + 1 == 0 (mod p) for prime p = 692881373, on September 6 2009. - Seiichi Manyama, Dec 30 2017
a(5) > 8*10^7. - Seiichi Manyama, Jan 01 2018
A superset of A193855. - Jud McCranie, Nov 06 2020

			tau(11) = 534612 and 11 | (534612 - 1), so a(1) = 11.
tau(23) = 18643272 and 23 | (18643272 - 1), so a(2) = 23.
tau(691) = -2747313442193908 and 691 | (-2747313442193908 - 1), so a(3) = 691.
tau(5807) = 237456233554906855056 and 5807 | (237456233554906855056 + 1), so a(4) = 5807.

N. Lygeros and O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.
Wikipedia, Ramanujan tau function.

Cf. A000594, A076847 (tau(p)), A007659, A193855, A273650, A295654.

Select[Prime@ Range[10^3], Function[p, AnyTrue[RamanujanTau[p] + {-1, 1}, Divisible[#, p] &]]] (* Michael De Vlieger, Dec 30 2017 *)

isok(p) = my(rp=ramanujantau(p)); isprime(p) && !((rp-1) % p) || !((rp+1) % p); \\ Michel Marcus, Nov 07 2020

A262339 Exceptional primes for Ramanujan's tau function.

Original entry on oeis.org

2, 3, 5, 7, 23, 691

Offset: 1

Jonathan Sondow, Sep 18 2015

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

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

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.

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

Cf. A000594, A046694, A193855.

A296580 Odd primes p such that tau(p) is congruent to (p-1)/2 (mod p), where tau is the Ramanujan tau function (A000594).

Original entry on oeis.org

191, 5399, 1259393

Offset: 1

Seiichi Manyama, Dec 16 2017

a(4) > 10^7.

There is no odd prime p (< 10^7) such that tau(p) is congruent to (p+1)/2 (mod p).

			tau(191) = 2762403350592 and 2762403350592 == 95 mod 191, so a(1) = 191.
tau(5399) = -616400667743946780600 and -616400667743946780600 == 2699 mod 5399, so a(2) = 5399.
tau(1259393) = -600367974333827988240021654527358 and -600367974333827988240021654527358 == 629696 mod 1259393, so a(3) = 1259393.

Eric Weisstein's World of Mathematics, Tau Function.
Wikipedia, Ramanujan tau function

Cf. A000594, A007659, A076847 (tau(p)), A193855, A273650, A273651, A295645.

cp's OEIS Frontend

A295645 Primes p such that tau(p) +- 1 is congruent to 0 (mod p), where tau is the Ramanujan tau function (A000594).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A262339 Exceptional primes for Ramanujan's tau function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A296580 Odd primes p such that tau(p) is congruent to (p-1)/2 (mod p), where tau is the Ramanujan tau function (A000594).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs