A300777 Primes of the form a^2 + b^2 such that both a^2 + 3b^2 and 3a^2 + b^2 are prime.

5, 29, 53, 113, 449, 653, 701, 809, 953, 1373, 1481, 1709, 1733, 2069, 2549, 2657, 2753, 2909, 3413, 3929, 4397, 5153, 5273, 5693, 6101, 6269, 6449, 6869, 7121, 7529, 7541, 7949, 8297, 8369, 8597, 8849, 9221, 9377, 9629, 10061, 10301, 10313, 10529, 10889, 10973, 11657, 12161, 12401, 12497, 12569

Offset

1,1

Comments

Primes of the form (x^2+y^2)/2 such that both x^2+xy+y^2 and x^2-xy+y^2 are prime.

Note that a^2+b^2 = ((a+b)^2+(a-b)^2)/2.

Links

Table of n, a(n) for n=1..50.

Formula

a(n) == 1 (mod 4).

Example

The prime 29 = 5^2 + 2^2 is a term, because 5^2 + 3*2^2 = 37 is prime and 3*5^2 + 2^2 = 79 is prime.
Equivalently: 29 = ((5+2)^2 + (5-2)^2)/2 = (7^2 + 3^2)/2 is a term, because 7^2 + 7*3 + 3^2 = 79 is prime and 7^2 - 7*3 + 3^2 = 37 is prime.

Maple

N:= 10^5: Res:= NULL:
for a from 1 to isqrt(N) by 2 do
for b from 2 to floor(sqrt(N-a^2)) by 2 do
   if isprime(a^2+b^2) and isprime(a^2+3*b^2) and isprime(3*a^2+b^2)
     then Res:= Res, a^2+b^2
   fi
od od:
sort([Res]); # Robert Israel, Mar 12 2018

Mathematica

lst = {}; nmx = 120; Do[ If[ PrimeQ[a^2 + b^2] && PrimeQ[3a^2 + b^2] && PrimeQ[a^2 + 3b^2], AppendTo[lst, a^2 + b^2]], {a, nmx}, {b, a, nmx}]; Take[ Sort@ Flatten@ lst, 50] (* Robert G. Wilson v, Mar 12 2018 *)

Prog

(PARI) lista(nn) = {vres = []; forstep(a=1, sqrtint(nn), 2, forstep(b=2, sqrtint(nn-a^2), 2, if (isprime(a^2+b^2) && isprime(a^2+3*b^2) && isprime(3*a^2+b^2), vres = concat(vres, a^2+b^2)); ); ); vecsort(vres); } \\ Michel Marcus, Apr 25 2018, after Maple

Crossrefs

Subsequence of A002313.

Cf. A007645, A182312.

Sequence in context: A107003 A141374 A147153 * A177831 A115706 A031394

Adjacent sequences: A300774 A300775 A300776 * A300778 A300779 A300780

Keyword

nonn

Author

Thomas Ordowski, Mar 12 2018

Extensions

More terms from Robert Israel and Robert G. Wilson v, Mar 12 2018

Status

approved