

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.


2



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
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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(lima^2)), 2, if(isprime(t=a^2+b^2) && isprime(ta*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



