OFFSET
1,1
COMMENTS
From Katherine E. Stange, Feb 03 2010: (Start)
Equivalently, primes p such that the smallest extension of F_p containing the cube roots of unity also contains the 9th roots of unity.
Equivalently, the primes p for which, if p^t = 1 mod 3, then p^t = 1 mod 9.
Equivalently, primes congruent to +/-1 modulo 9.
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)
Primes in A056020. - Reinhard Zumkeller, Jan 07 2012
Primes congruent to (1,17) mod 18. - Vincenzo Librandi, Aug 14 2012
Equivalently, primes such that p^2 == 1 (mod 9). - M. F. Hasler, Apr 16 2022
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emma Lehmer, On special primes, Pac. J. Math., 118 (1985), 471-478.
J. H. Silverman and K. E. Stange. Amicable pairs and aliquot cycles for elliptic curves, arxiv:0912.1831 [math.NT], 2009.
MATHEMATICA
Union[Join[Select[Range[-1, 3000, 18], PrimeQ], Select[Range[1, 3000, 18], PrimeQ]]] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2012 *)
Select[Prime[Range[4000]], MemberQ[{1, 17}, Mod[#, 18]]&] (* Vincenzo Librandi, Aug 14 2012 *)
PROG
(Haskell)
a129805 n = a129805_list !! (n-1)
a129805_list = [x | x <- a056020_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012
(Magma) [ p: p in PrimesUpTo(1300) | p mod 18 in {1, 17} ]; // Vincenzo Librandi, Aug 14 2012
(PARI) select( {is_A129805(n)=n^2%9==1&&isprime(n)}, primes(199)) \\ M. F. Hasler, Apr 16 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 22 2007
STATUS
approved