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

1,1

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.

Prog

(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

Crossrefs

Cf. A002407, A275879.

Sequence in context: A141952 A289723 A064398 * A129079 A249556 A135165

Adjacent sequences: A275875 A275876 A275877 * A275879 A275880 A275881

Keyword

nonn

Author

N. J. A. Sloane, Aug 17 2016

Extensions

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

Status

approved