A002344 -id:A002344 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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)

Essentially the same as A045373. Primes in A020670.

Cf. A002344, A002345, A139643.

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

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

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

A002345 Numbers y such that p = x^2 + 7y^2, with prime p = A033207(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 1, 3, 1, 2, 4, 3, 4, 2, 1, 3, 5, 5, 4, 2, 5, 4, 5, 1, 6, 4, 2, 1, 4, 7, 7, 6, 7, 2, 4, 6, 5, 7, 8, 8, 3, 7, 1, 5, 6, 3, 2, 8, 9, 7, 6, 8, 9, 4, 2, 5, 8, 1, 10, 10, 3, 7, 5, 2, 8, 10, 9, 7

Offset: 1

N. J. A. Sloane

nonn

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

Cf. A002344.

cp's OEIS Frontend

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

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