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A300845
a(n) is the smallest prime q such that q^2 + q*p + p^2 is a prime number where p is n-th prime, or 0 if no such q exists.
1
3, 2, 7, 2, 3, 2, 3, 11, 3, 3, 3, 2, 7, 3, 19, 7, 7, 2, 11, 13, 2, 5, 37, 19, 11, 3, 5, 3, 5, 13, 3, 7, 7, 2, 7, 5, 2, 3, 37, 7, 3, 29, 13, 5, 3, 11, 17, 29, 37, 2, 13, 3, 2, 67, 19, 7, 7, 5, 3, 3, 29, 43, 23, 7, 5, 3, 3, 5, 7, 2, 43, 3, 2, 17, 17, 7, 19, 2, 13, 23, 43, 3, 7, 2, 2, 7, 7, 2, 7
OFFSET
1,1
COMMENTS
Probably, for each prime p, there is prime q such that q^2 + q*p + p^2 is also a prime since the existence of such q is a special case of Hypothesis H of Schinzel and Sierpinski. However, this is not proven yet.
Corresponding generalized cuban primes are 19, 19, 109, 67, 163, 199, 349, 691, 607, 937, 1063, 1447, 2017, 1987, 3463, 3229, 3943, 3847, 5347, 6133, ...
EXAMPLE
a(3) = 7 because 7^2 + 7*5 + 5^2 = 109 is prime number and 7 is the least prime with this property.
MAPLE
f:= proc(p) local q;
q:= 1;
do
q:= nextprime(q);
if isprime(q^2+q*p+p^2) then return q fi;
od
end proc:
map(f, select(isprime, [2, seq(i, i=3..1000, 2)])); # Robert Israel, Mar 13 2018
MATHEMATICA
Table[Block[{q = 2}, While[! PrimeQ[q^2 + q p + p^2], q = NextPrime@ q]; q], {p, Prime@ Range@ 89}] (* Michael De Vlieger, Mar 14 2018 *)
PROG
(PARI) a(n) = {my(p=prime(n)); forprime(q=2, , if(isprime(p^2+p*q+q^2), return(q)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Mar 13 2018
STATUS
approved