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

Author

Giovanni Resta, Dec 12 2007

Keywords

• nonn

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)

Examples

			tau(63001) = -80561663527802406257321747 which is prime.
		

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

Crossrefs

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

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

Programs

Mathematica

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

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