A266966 - OEIS

A266966

Primes of the form x^2 + y^2 where x^2 + x*y + y^2 is also prime with positive x and y.

2, 5, 13, 37, 41, 53, 73, 109, 137, 157, 173, 181, 193, 197, 233, 349, 373, 401, 421, 457, 509, 541, 557, 569, 577, 613, 661, 709, 733, 757, 769, 821, 877, 941, 1009, 1033, 1069, 1117, 1129, 1193, 1201, 1237, 1301, 1373, 1453, 1493, 1549, 1597, 1621, 1657, 1669, 1697, 1721

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OFFSET

1,1

COMMENTS

Sequence focuses on the positive values of x and y. Otherwise, for example 17 = 4^2 + (-1)^2 and 4^2 + 4*(-1) + (-1)^2 = 13 is prime too.

Corresponding generalized cuban primes are 3, 7, 19, 43, 61, 67, 97, 139, 181, 199, 211, 223, 271, 277, 337, 421, 439, 499, 541, 601, 619, 631, 751, 787, 811, 823, 829, 919, ...

LINKS

Table of n, a(n) for n=1..53.

EXAMPLE

5 is a term because 2^2 + 1^2 = 5 is prime and 2^2 + 2*1 + 1^2 = 7 is prime.

13 is a term because 3^2 + 2^2 = 13 is prime and 3^2 + 3*2 + 2^2 = 19 is prime.

37 is a term because 6^2 + 1^2 = 37 is prime and 6^2 + 6*1 + 1^2 = 43 is prime.

MATHEMATICA

lim = 50; Take[Select[Union@ Flatten@ Table[ If[PrimeQ[Abs[x^2 + x y + y^2]], x^2 + y^2, Nothing], {x, lim}, {y, lim}], PrimeQ], 53] (* Michael De Vlieger, Jan 07 2016 *)

PROG

(PARI) list(lim) = my(v=List(), t); lim\=1; for(x=1, sqrtint(lim), for(y=1, min(sqrtint(lim-x^2), x), if(isprime(t=x^2+y^2) && isprime(x^2+x*y+y^2), listput(v, t)))); vecsort(Vec(v), , 8)

CROSSREFS

Cf. A002313, A007645.

Sequence in context: A148299 A148300 A038982 * A019415 A262203 A175118

Adjacent sequences: A266963 A266964 A266965 * A266967 A266968 A266969

KEYWORD

nonn

AUTHOR

Altug Alkan, Jan 07 2016

STATUS

approved