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%I #46 Sep 08 2022 08:45:30
%S 17,19,37,53,71,73,89,107,109,127,163,179,181,197,199,233,251,269,271,
%T 307,359,379,397,431,433,449,467,487,503,521,523,541,557,577,593,613,
%U 631,647,683,701,719,739,757,773,809,811,827,829,863,881,883,919,937,953,971,991
%N Primes congruent to +-1 mod 18.
%C From _Katherine E. Stange_, Feb 03 2010: (Start)
%C Equivalently, primes p such that the smallest extension of F_p containing the cube roots of unity also contains the 9th roots of unity.
%C Equivalently, the primes p for which, if p^t = 1 mod 3, then p^t = 1 mod 9.
%C Equivalently, primes congruent to +/-1 modulo 9.
%C Membership or non-membership of the prime p in this sequence and sequence A002144 (primes congruent to 1 mod 4; equivalently, primes p such that the smallest extension of F_p containing the square roots of unity contains the 4th roots of unity) appear to determine the behavior of amicable pairs on the elliptic curve y^2 = x^3 + p (Silverman, Stange 2009). (End)
%C Primes in A056020. - _Reinhard Zumkeller_, Jan 07 2012
%C Primes congruent to (1,17) mod 18. - _Vincenzo Librandi_, Aug 14 2012
%C Equivalently, primes such that p^2 == 1 (mod 9). - _M. F. Hasler_, Apr 16 2022
%H Reinhard Zumkeller, <a href="/A129805/b129805.txt">Table of n, a(n) for n = 1..10000</a>
%H Emma Lehmer, <a href="https://projecteuclid.org/euclid.pjm/1102706454">On special primes</a>, Pac. J. Math., 118 (1985), 471-478.
%H J. H. Silverman and K. E. Stange. <a href="http://arxiv.org/abs/0912.1831">Amicable pairs and aliquot cycles for elliptic curves</a>, arxiv:0912.1831 [math.NT], 2009.
%t Union[Join[Select[Range[-1, 3000, 18], PrimeQ], Select[Range[1, 3000, 18], PrimeQ]]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 18 2012 *)
%t Select[Prime[Range[4000]],MemberQ[{1,17},Mod[#,18]]&] (* _Vincenzo Librandi_, Aug 14 2012 *)
%o (Haskell)
%o a129805 n = a129805_list !! (n-1)
%o a129805_list = [x | x <- a056020_list, a010051 x == 1]
%o -- _Reinhard Zumkeller_, Jan 07 2012
%o (Magma) [ p: p in PrimesUpTo(1300) | p mod 18 in {1, 17} ]; // _Vincenzo Librandi_, Aug 14 2012
%o (PARI) select( {is_A129805(n)=n^2%9==1&&isprime(n)}, primes(199)) \\ _M. F. Hasler_, Apr 16 2022
%Y Cf. A000040, A010051.
%Y Cf. A129806, A129807.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, May 22 2007
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