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%I #15 Aug 04 2021 02:19:44
%S 3,79,997,2657,3697,4513,6947,8887,9547,16187,22697,26479,31319,37463,
%T 39139,39887,43573,43987,45667,47387,47743,47819,48221,54217,56923,
%U 57373,74017,74149,74707,75533,93251,100043
%N Primes p with prime(p)^3 + 2*p^3 and p^3 + 2*prime(p)^3 both prime.
%C Conjecture: This sequence has infinitely many terms.
%C In 2001 Heath-Brown proved that there are infinitely many primes of the form x^3 + 2*y^3 with x and y positive integers.
%H Zhi-Wei Sun, <a href="/A236574/b236574.txt">Table of n, a(n) for n = 1..10000</a>
%H D. R. Heath-Brown, <a href="https://doi.org/10.1007/BF02392715">Primes represented by x^3 + 2y^3</a>. Acta Mathematica 186 (2001), pp. 1-84.
%e a(1) = 3 since prime(3)^3 + 2*3^3 = 125 + 54 = 179 and 3^3 + 2*prime(3)^3 = 27 + 2*125 = 277 are both prime, but 2^3 + 2*prime(2)^3 = 62 is composite.
%t p[n_]:=PrimeQ[Prime[n]^3+2*n^3]&&PrimeQ[n^3+2*Prime[n]^3]
%t n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]
%t Select[Prime[Range[10000]],AllTrue[{Prime[#]^3+2*#^3,#^3+2*Prime[ #]^3}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Aug 20 2017 *)
%Y Cf. A000040, A000578, A173587, A220413, A236193.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jan 29 2014
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