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Number of ways to write n as p^2 + q with p and q both prime.
7

%I #18 Feb 05 2017 13:16:24

%S 0,0,0,0,0,1,1,0,1,0,2,1,0,1,1,1,1,0,0,1,1,1,1,0,0,1,2,2,0,1,0,2,1,0,

%T 1,1,0,2,0,1,1,1,0,1,1,1,1,1,0,1,2,2,0,2,0,3,1,0,0,1,0,3,1,0,1,2,0,3,

%U 0,1,1,2,0,0,1,1,1,2,0,2,0,1,1,1,0,2,1,1,0,1,0,3,1,0,0,2,0,2,0,0

%N Number of ways to write n as p^2 + q with p and q both prime.

%C Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5. Moreover, if n = 6*k + 2, then a(n) > 0 except for k = 0, 1, 12, 28, 102, 117, 168, 4079.

%C We have verified the conjecture for n up to 10^9.

%H Zhi-Wei Sun, <a href="/A258139/b258139.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: (Sum_{k>=1} x^prime(k))*(Sum_{k>=1} x^(prime(k)^2)). - _Ilya Gutkovskiy_, Feb 05 2017

%e a(11) = 2 since 11 = 2^2 + 7 = 3^2 + 2 with 2, 3, 7 all prime.

%t Do[r=0;Do[If[PrimeQ[n-Prime[k]^2],r=r+1],{k,1,PrimePi[Sqrt[n]]}];Print[n," ",r];Continue,{n,1,100}]

%Y Cf. A000040, A002375, A258140, A258141.

%K nonn

%O 1,11

%A _Zhi-Wei Sun_, May 22 2015