A135430 - OEIS

%I #52 Jun 05 2024 13:36:51

%S 63001,458329,942841,966289,1510441,2961841,4879681,14280841,29019769,

%T 46117681,49182169,51652969,56957209,75047569,80120401,86136961,

%U 93644329,97752769,104509729,162384049,164378041,177235969,193571569

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

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

%C 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.

%C From _Olivier Rozier_, Feb 03 2016 (Start)

%C 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.

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

%H Dana Jacobsen, <a href="/A135430/b135430.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Bennett, Adela Gherga, Vandita Patel, and Samir Siksek, <a href="https://arxiv.org/abs/2101.02933">Odd values of the Ramanujan tau function</a>, arXiv:2101.02933 [math.NT], 2021.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2313305">The Primality of Ramanujan's Tau-Function</a>, The American Mathematical Monthly, Vol. 72, No. 2, Part 2 (Feb., 1965), pp. 15-18.

%H N. Lygeros and O. Rozier, <a href="http://dx.doi.org/10.1007/s11139-012-9420-8">Odd prime values of the Ramanujan tau function</a>, Ramanujan Journal, Vol. 32 (2013), pp. 269-280.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TauFunctionPrime.html">Tau Function Prime</a>

%e tau(63001) = -80561663527802406257321747 which is prime.

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

%o (PARI) for(x=1,1000, n=(2*x+1)^2; if(isprime(abs(ramanujantau(n))), print1(n", "))) \\ _Dana Jacobsen_, Sep 07 2015

%o (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

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

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

%K nonn

%O 1,1

%A _Giovanni Resta_, Dec 12 2007