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A266966
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Primes of the form x^2 + y^2 where x^2 + x*y + y^2 is also prime with positive x and y.
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0
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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
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1,1
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COMMENTS
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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, ...
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LINKS
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Table of n, a(n) for n=1..53.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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Cf. A002313, A007645.
Sequence in context: A148299 A148300 A038982 * A019415 A262203 A175118
Adjacent sequences: A266963 A266964 A266965 * A266967 A266968 A266969
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KEYWORD
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nonn
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AUTHOR
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Altug Alkan, Jan 07 2016
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STATUS
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approved
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