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

%I A135430 #53 Feb 16 2025 08:33:07
%S A135430 63001,458329,942841,966289,1510441,2961841,4879681,14280841,29019769,
%T A135430 46117681,49182169,51652969,56957209,75047569,80120401,86136961,
%U A135430 93644329,97752769,104509729,162384049,164378041,177235969,193571569
%N A135430 Numbers k for which Ramanujan's function tau(k)=A000594(k) is an odd prime.
%C A135430 Here, negative integers whose absolute value is prime are considered prime.
%C A135430 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 A135430 From _Olivier Rozier_, Feb 03 2016 (Start)
%C A135430 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 A135430 It is conjectured that the equation |tau(n)|=2 has no solution. (End)
%H A135430 Dana Jacobsen, <a href="/A135430/b135430.txt">Table of n, a(n) for n = 1..1000</a>
%H A135430 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 A135430 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 A135430 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 A135430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunctionPrime.html">Tau Function Prime</a>
%e A135430 tau(63001) = -80561663527802406257321747 which is prime.
%t A135430 Select[Range[1, 7000, 2]^2, PrimeQ@RamanujanTau@# &]
%o A135430 (PARI) for(x=1,1000, n=(2*x+1)^2; if(isprime(abs(ramanujantau(n))), print1(n", "))) \\ _Dana Jacobsen_, Sep 07 2015
%o A135430 (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 A135430 Cf. A000594 (Ramanujan's tau function tau(n)).
%Y A135430 Cf. A265913 (tau(a(n))).
%K A135430 nonn
%O A135430 1,1
%A A135430 _Giovanni Resta_, Dec 12 2007