A135430 Numbers k for which Ramanujan's function tau(k)=A000594(k) is an odd prime.

63001, 458329, 942841, 966289, 1510441, 2961841, 4879681, 14280841, 29019769, 46117681, 49182169, 51652969, 56957209, 75047569, 80120401, 86136961, 93644329, 97752769, 104509729, 162384049, 164378041, 177235969, 193571569

Offset

1,1

Comments

Here, negative integers whose absolute value is prime are considered prime.

a(1) = 63001 was found by Lehmer in 1965. It is known that tau(n) is odd if and only if n is an odd square. Indeed, a(1)=251^2, a(2)=677^2, ..., a(7)=47^4. The first sixth power in the sequence is 1151^6.

From Olivier Rozier, Feb 03 2016 (Start)

a(n) = p^(q-1) for p,q odd primes, and p not included in A007659, so that a(n) is a subsequence of A036454. Consequence of the arithmetical properties: (i) tau function is multiplicative, (ii) for p prime, tau(p^(k-1)) is the k-th term of a Lucas sequence.

It is conjectured that the equation |tau(n)|=2 has no solution. (End)

Links

Dana Jacobsen, Table of n, a(n) for n = 1..1000

Michael Bennett, Adela Gherga, Vandita Patel, and Samir Siksek, Odd values of the Ramanujan tau function, arXiv:2101.02933 [math.NT], 2021.

D. H. Lehmer, The Primality of Ramanujan's Tau-Function, The American Mathematical Monthly, Vol. 72, No. 2, Part 2 (Feb., 1965), pp. 15-18.

N. Lygeros and O. Rozier, Odd prime values of the Ramanujan tau function, Ramanujan Journal, Vol. 32 (2013), pp. 269-280.

Eric Weisstein's World of Mathematics, Tau Function Prime

Example

tau(63001) = -80561663527802406257321747 which is prime.

Mathematica

Select[Range[1, 7000, 2]^2, PrimeQ@RamanujanTau@# &]

Prog

(PARI) for(x=1, 1000, n=(2*x+1)^2; if(isprime(abs(ramanujantau(n))), print1(n", "))) \\ Dana Jacobsen, Sep 07 2015
(Perl) use ntheory ":all"; for (0..1000) { my $n = (2*$_+1)**2; say $n if is_prime(abs(ramanujan_tau($n))); } # Dana Jacobsen, Sep 07 2015

Crossrefs

Cf. A000594 (Ramanujan's tau function tau(n)).

Cf. A265913 (tau(a(n))).

Sequence in context: A224649 A253854 A237675 * A049025 A322679 A322689

Adjacent sequences: A135427 A135428 A135429 * A135431 A135432 A135433

Keyword

nonn

Author

Giovanni Resta, Dec 12 2007

Status

approved