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A193855 Primes p such that tau(p) is congruent to 1 (mod p), where tau is the Ramanujan tau function. 3
11, 23, 691 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

M. J. Hopkins wrote "It is not known whether or not tau(p) == 1 mod p holds for infinitely many primes". For more information about this open problem see the Sloane comment in A000594.

a(4) > 500000. - Dana Jacobsen, Sep 06 2015

a(4) > 10^7. - Seiichi Manyama, Nov 25 2017

Terms 23 and 691 are exceptional primes for Ramanujan's tau function, see A262339. - Jud McCranie, Nov 05 2020

A subset of A295645. - Jud McCranie, Nov 06 2020

REFERENCES

M. J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317.

M. J. Hopkins, Algebraic topology and modular forms, ICM 2002, Vol. I, pp. 283-309.

LINKS

Table of n, a(n) for n=1..3.

M. J. Hopkins, Algebraic topology and modular forms, arXiv:math/0212397 [math.AT], 2002.

B. Mazur and A. Wiles, On p-adic analytic families of Galois representations, Compositio Mathematica, tome 59, n. 2 (1986), p. 231-264.

MATHEMATICA

Select[Prime[Range[1, 1000]], 1 == Mod[RamanujanTau[#], #] &] (* Robert Price, May 20 2015 *)

PROG

(Perl) use ntheory ":all"; forprimes { say if (ramanujan_tau($_) % $_) == 1; } 1000; # Dana Jacobsen, Sep 06 2015

(PARI 2.8) forprime(n=1, 1000, if(Mod(tauramanujan(n), n)==1, print1(n, ", "))) \\ Dana Jacobsen, Sep 06 2015

CROSSREFS

Cf. A000594, A262339, A295645.

Sequence in context: A153318 A005485 A041240 * A295645 A295654 A247347

Adjacent sequences:  A193852 A193853 A193854 * A193856 A193857 A193858

KEYWORD

nonn,bref,hard,more

AUTHOR

Omar E. Pol, Aug 14 2011

STATUS

approved

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Last modified January 29 01:19 EST 2022. Contains 350670 sequences. (Running on oeis4.)