A033207 Primes of the form x^2 + 7*y^2.

7, 11, 23, 29, 37, 43, 53, 67, 71, 79, 107, 109, 113, 127, 137, 149, 151, 163, 179, 191, 193, 197, 211, 233, 239, 263, 277, 281, 317, 331, 337, 347, 359, 373, 379, 389, 401, 421, 431, 443, 449, 457, 463, 487, 491

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Except for a(1) = 7, these are the primes which can be written in the form a^2 + 7*b^2 with a > 0 and b > 0. - V. Raman, Sep 08 2012

These are the primes p for which p^3 - 1 is divisible by 7, with two exceptions: p = 2 is not in the sequence even though 2^3 - 1 is divisible by 7, and p = 7 is in the sequence even though 7^3 - 1 is not divisible by 7. Except for p = 7, if p^3 - 1 is not divisible by 7, it is congruent to 5 (mod 7). - Richard R. Forberg, Jun 03 2013

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Sushma Palimar and B. R. Shankar, Mersenne Primes in Real Quadratic Fields, Journal of Integer Sequences, Vol. 15 (2012), #12.5.6.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Essentially the same as A045373. Primes in A020670.

Cf. A002344, A002345, A139643.

Programs

Mathematica

QuadPrimes2[1, 0, 7, 10000] (* see A106856 *)

PARI

is(n)=kronecker(n,7)>=0 && isprime(n) && n>2 \\ Charles R Greathouse IV, Nov 19 2012

Formula

Primes congruent to {1, 7, 9, 11, 15, 23, 25} (mod 28). - T. D. Noe, Apr 29 2008