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.

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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}]

Crossrefs

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

Sequence in context: A306518 A333310 A049986 * A137289 A211359 A211357

Adjacent sequences: A218794 A218795 A218796 * A218798 A218799 A218800

Keyword

nonn

Author

Zhi-Wei Sun, Nov 05 2012

Status

approved