A192056 - OEIS

%I #10 Dec 30 2012 14:18:40

%S 0,0,1,1,1,2,2,2,2,1,3,1,2,2,3,3,2,1,3,4,2,6,2,1,8,3,3,6,2,1,3,3,1,5,

%T 7,5,4,4,3,3,6,3,3,6,3,5,3,7,5,7,6,4,5,1,8,8,2,4,6,1,5,2,4,9,8,3,6,7,

%U 3,5,5,5,3,3,5,9,4,13,6,5,9,7,7,3,10,9,8,9,7,4,7,13,5,7,10,4,4,11,4,5

%N a(n) = |{0<k<n/2: k^2+(n-k)^2-3(n-1 mod 2) is prime and JacobiSymbol[k,n+3(n-1 mod 2)]=1}|

%C Conjecture: a(n)>0 for all n>2. Moreover, if n>2 is not among 12, 18, 105, 522, then there is 0<k<n/2 such that

%C k^2+(n-k)^2-3(n-1 mod 2) is prime and also JacobiSymbol[k,p]=1 for any prime divisor p of n+3(n-1 mod 2).

%C This conjecture has been verified for n up to 2*10^8. It is stronger than Ming-Zhi Zhang's conjecture that any odd integer n>1 can be written as x+y (x,y>0) with x^2+y^2 prime (see A036468).

%H Zhi-Wei Sun, <a href="/A192056/b192056.txt">Table of n, a(n) for n = 1..10000</a>

%H  Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.

%e a(33)=1 since 4^2+29^2=857 is prime and JacobiSymbol[4,33]=1.

%e a(24)=1 since 10^2+14^2-3=293 is prime and JacobiSymbol[10,24+3]=1.

%t a[n_]:=a[n]=Sum[If[PrimeQ[k^2+(n-k)^2-3Mod[n-1,2]]==True&&JacobiSymbol[k,n+3Mod[n-1,2]]==1,1,0],{k,1,(n-1)/2}]

%t Do[Print[n," ",a[n]],{n,1,100}]

%Y Cf. A036468, A191004, A185150, A220554.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Dec 30 2012