A218797 - OEIS

%I #14 Nov 06 2012 11:12:40

%S 0,0,0,1,0,1,2,1,2,0,1,2,2,1,3,2,3,1,2,1,1,2,2,2,1,2,1,2,4,1,3,2,2,2,

%T 2,2,2,4,3,3,3,2,4,4,3,0,2,1,1,1,1,2,2,3,2,4,4,3,3,2,3,4,2,2,3,2,1,3,

%U 3,1,2,2,5,1,4,2,2,1,1,6,3,1,5,1,1,5,4,1,4,1,2,6,2,4,2,2,2,1,4,4

%N Number of ways to write 2n - 1 as p + q + r with p <= q <= r and p, q, r, p^2 + q^2 + r^2 all prime.

%C Conjecture: a(n) > 0 for all n=1715,1716,....

%C This conjecture is stronger than the weak Goldbach conjecture. It has been verified for n up to 500,000. Those 0<n<1715 with a(n)=0 are 1, 2, 3, 5, 10, 46, 126, 129, 154, 201, 385, 426, 475, 1714.

%H Zhi-Wei Sun, <a href="/A218797/b218797.txt">Table of n, a(n) for n = 1..10000</a>

%e a(7)=2 since 13=3+3+7=3+5+5, and both 3^2+3^2+7^2=67 and 3^2+5^2+5^2=59 are primes.

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

%t Do[Print[n," ",a[2n-1]],{n,1,10000}]

%Y Cf. A000040, A054860, A085317, A218754, A218585, A218654, A218656.

%K nonn

%O 1,7

%A _Zhi-Wei Sun_, Nov 05 2012