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A218797
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.
2
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, 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, 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
OFFSET
1,7
COMMENTS
Conjecture: a(n) > 0 for all n=1715,1716,....
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.
EXAMPLE
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.
MATHEMATICA
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]}]
Do[Print[n, " ", a[2n-1]], {n, 1, 10000}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 05 2012
STATUS
approved