A129806 -id:A129806 - cp's OEIS frontend

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

N. J. A. Sloane, May 22 2007

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

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.

Cf. A000040, A010051.

Cf. A129806, A129807.

a129805 n = a129805_list !! (n-1)
a129805_list = [x | x <- a056020_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[ p: p in PrimesUpTo(1300) | p mod 18 in {1, 17} ]; // Vincenzo Librandi, Aug 14 2012

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 *)

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

A129807 Primes congruent to +-7 mod 18.

Original entry on oeis.org

7, 11, 29, 43, 47, 61, 79, 83, 97, 101, 137, 151, 173, 191, 223, 227, 241, 263, 277, 281, 313, 317, 331, 349, 353, 367, 389, 421, 439, 443, 457, 461, 479, 547, 569, 587, 601, 619, 641, 659, 673, 677, 691, 709, 727, 821, 839, 853, 857, 907, 911, 929, 947, 983, 997, 1019

Offset: 1

N. J. A. Sloane, May 22 2007

Also: primes that are sums of three consecutive terms of A001651. These sum to either 3k+1+3k+2+3k+4=9k+7, candidates for A061241, or 3k+2+3k+4+3k+5=9k+11, candidates for A061238. - R. J. Mathar, Jun 10 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Emma Lehmer, On special primes, Pac. J. Math., 118 (1985), 471-478.

Cf. A000040.
Cf. A001651, A061238, A061241.
Cf. A129805, A129806.

[ p: p in PrimesUpTo(1300) | p mod 18 in {7, 11} ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{7,11},Mod[ #,18]]&] (* Zak Seidov, May 23 2007 *)

Conjecture: Equals (A061241 UNION A061238) MINUS {2}. - R. J. Mathar, Jun 10 2007

A154291 Primes of the form 4x^3 + 27y^2, with x<0.

Original entry on oeis.org

23, 31, 139, 211, 239, 419, 491, 499, 563, 643, 743, 751, 823, 1291, 1319, 1427, 1931, 2039, 2687, 2767, 3011, 3119, 3163, 3191, 3299, 3307, 3803, 3919, 4027, 4091, 4099, 4423, 4703, 4999, 5323, 5639, 5647, 6007, 6043, 6079, 6323, 6691, 6719, 6763, 7331

Offset: 1

T. D. Noe, Jan 06 2009, Jun 18 2009, Jun 21 2009

nonn

For each prime p, the elliptic curve 27y^2 = 4x^3 + p must be solved to determine whether there is an integer solution with x positive. About 2/3 of all primes can be eliminated because p-4x^3 is never divisible by 27. The remaining primes are congruent to +-5 (mod 18). Hence this sequence is a subsequence of A129806. Half of those primes can be eliminated because even when 27 does divide p-4x^3, the quotient must equal 1 (mod 4) in order to be a square. Hence all these primes must equal 23 or 31 (mod 36). James Buddenhagen used APECS and I used Sage to examine the elliptic curves. The first difficult prime is 1831. All the elliptic curves with p = 23 or 31 (mod 36) appear to have trivial torsion and rank 0 or 2.

See the link to the Sage/Python program to see how the problem with 1831 was resolved. The first prime producing an elliptic curve of rank 4 is 19427.

			743 = 4*(-17977)^3 + 27*927735^2

T. D. Noe, Sage/Python program

Cf. A153635, A153636

cp's OEIS Frontend

A129805 Primes congruent to +-1 mod 18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

A129807 Primes congruent to +-7 mod 18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A154291 Primes of the form 4x^3 + 27y^2, with x<0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs