A129805 Primes congruent to +-1 mod 18.

17, 19, 37, 53, 71, 73, 89, 107, 109, 127, 163, 179, 181, 197, 199, 233, 251, 269, 271, 307, 359, 379, 397, 431, 433, 449, 467, 487, 503, 521, 523, 541, 557, 577, 593, 613, 631, 647, 683, 701, 719, 739, 757, 773, 809, 811, 827, 829, 863, 881, 883, 919, 937, 953, 971, 991

Offset: 1

Author

N. J. A. Sloane, May 22 2007

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000040, A010051.

Cf. A129806, A129807.

Programs

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

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 *)

PARI

select( {is_A129805(n)=n^2%9==1&&isprime(n)}, primes(199)) \\ M. F. Hasler, Apr 16 2022