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%I #30 Aug 04 2021 02:19:05
%S 0,1,0,0,2,2,1,2,2,2,2,1,2,1,1,1,1,2,2,2,2,3,1,2,1,1,2,4,3,4,2,1,2,2,
%T 2,1,1,2,3,2,2,4,3,3,5,4,3,3,1,4,3,2,2,2,3,2,1,3,3,4,3,7,2,5,2,3,5,5,
%U 5,4,3,2,3,2,3,5,2,2,4,5,4,4,2,4,9,4,6,7,5,3,3,4,3,3,9,5,3,3,3,5
%N a(n) = |{0<k<n: 2*n+k and 2*n^3+k^3 are both prime}|.
%C Conjecture: a(n)>0 for all n>4.
%C This has been verified for n up to 10^8.
%C We also conjecture that for any integer n>1 there is an integer 0<k<n such that n^2+k^2 is prime.
%H Zhi-Wei Sun, <a href="/A224030/b224030.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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%e a(7) = 1 since 2*7+5 = 19 and 2*7^3+5^3 = 811 are both prime.
%e a(57) = 1 since 2*57+23 = 137 and 2*57^3+23^3 = 382553 are both prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[2n+k]==True&&PrimeQ[2n^3+k^3]==True,1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A185636, A204065, A220413.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Apr 15 2013
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