A182312 Primes of the form a^2 + b^2 such that both a^2 + b^2 - a*b and a^2 + b^2 + a*b are prime.

5, 13, 37, 109, 193, 421, 457, 541, 613, 709, 757, 1033, 1117, 1201, 1549, 1597, 1621, 1789, 2137, 2293, 2377, 2437, 2797, 3061, 3109, 3313, 3361, 3469, 4153, 4621, 4657, 4729, 5077, 5233, 5569, 5653, 6421, 6469, 6637, 6997, 7417, 7561, 7681, 7753, 8101, 8689

Offset

1,1

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) == 1 (mod 4). - Thomas Ordowski, Mar 13 2018

Example

The prime 13 = 2^2 + 3^2 is a term, since 13 - 2*3 = 7 is prime and 13 + 2*3 = 19 is prime.

Mathematica

prsQ[{a_, b_}]:=Module[{c=a^2+b^2, d=a*b}, And@@PrimeQ[c+{0, d, -d}]]; Sort[#[[1]]^2+#[[2]]^2&/@Select[Subsets[Range[100], {2}], prsQ]] (* Harvey P. Dale, Apr 27 2014 *)

Prog

(PARI) list(lim)=my(v=List(), t); for(a=1, sqrt(lim), forstep(b=1+a%2, min(a, sqrt(lim-a^2)), 2, if(isprime(t=a^2+b^2) && isprime(t-a*b) && isprime(t+a*b), listput(v, t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 25 2012

Crossrefs

Subsequence of A002313.

Cf. A007645.

Sequence in context: A298417 A193642 A220709 * A071100 A199108 A125734

Adjacent sequences: A182309 A182310 A182311 * A182313 A182314 A182315

Keyword

nonn

Author

Thomas Ordowski, Apr 24 2012

Extensions

a(6)-a(46) from Charles R Greathouse IV, Apr 25 2012

Status

approved