A275878 Standard Jacobi primes.

7, 61, 331, 547, 1951, 2437, 3571, 4219, 7351, 8269, 9241, 10267, 13669, 23497, 25117, 55897, 60919, 74419, 89269, 92401, 102121, 112327, 115837, 126691, 145861, 170647, 202021, 231019, 241117, 246247, 251431, 267307, 283669, 329677, 347821, 360187, 372769

Offset: 1

Author

N. J. A. Sloane, Aug 17 2016

Keywords

• nonn

Comments

From Peter Bala, Feb 20 2022: (Start)

Primes of the form (3*k + 2)^3 - (3*k + 1)^3 = 27*k^2 + 27*k + 7.

Equivalently, primes p such that 4*p = 27*x^2 + 1, where x is odd.

Primes p of the form 6*m + 1, where 8*m + 1 is an odd square.

A prime p is in this list iff binomial(2*(p-1)/3,(p-1)/3) == -1 (mod p). See Cosgrave and Dilcher, Theorem 5, Corollary 3. (End)

Subsequence of cuban primes (A002407). - Bernard Schott, Jul 28 2022

Links

• Dana Jacobsen, Table of n, a(n) for n = 1..10000
• John B. Cosgrave and Karl Dilcher, An Introduction to Gauss Factorials, Amer. Math. Monthly, Vol. 118, No. 9 (November 2011), pp. 812-829.
• J. B. Cosgrave and Karl Dilcher, A role for generalized Fermat numbers, Math. Comp., to appear 2016; see also Paper #10, See Table 7.1.

Crossrefs

Cf. A002407, A275879.

Programs

Perl

use ntheory ":all"; forprimes { if (($%3)==1) { my $z = znorder(factorial(($-1)/3),$); $z/=3 unless $z%3; say if $z==1; } } 1e6; # _Dana Jacobsen, Aug 18 2016

Perl

use ntheory ":all"; for (0..1000) { my $p = 27*$*$ + 27*$ + 7; say $p if is_prime($p); } # _Dana Jacobsen, Aug 18 2016

Extensions

Terms a(21) and beyond from Dana Jacobsen, Aug 18 2016