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A230451
Number of ways to write n = x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 3, 2*z + 5 are all prime and x*y*z is a triangular number.
3
0, 0, 1, 0, 2, 3, 0, 4, 3, 1, 7, 3, 2, 3, 7, 4, 5, 6, 3, 4, 8, 5, 8, 3, 6, 8, 9, 9, 5, 12, 2, 11, 4, 4, 4, 13, 5, 9, 13, 8, 14, 8, 3, 15, 7, 8, 10, 6, 5, 17, 15, 4, 6, 9, 8, 10, 15, 9, 7, 15, 11, 5, 6, 11, 14, 14, 7, 11, 3, 12, 23, 16, 5, 20, 14, 4, 9, 14, 5, 19, 19, 4, 3, 12, 7, 16, 5, 12, 6, 11, 12, 12, 23, 14, 23, 12, 5, 17, 14, 5
OFFSET
1,5
COMMENTS
Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 7.
(ii) Any integer n > 7 can be written as x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 1, 2*x*y + 1 are primes and x*y*z is a triangular number.
(iii) Each integer n > 4 not equal to 7 or 14 can be expressed as p + q + r (p, q, r > 0) with p and 2*q + 1 both primes, and p*q*r a triangular number.
(iv) Any integer n > 6 not among 16, 20, 60 can be written as x + y + z (x, y, z > 0) such that x*y + x*z + y*z is a triangular number.
Part (i) is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.
See also A227877 and A230596 for some related conjectures.
LINKS
EXAMPLE
a(6) = 3 since 6 = 1 + 2 + 3 = 2 + 1 + 3 = 3 + 2 + 1, and 2*1 + 1 = 3, 2*2 + 3 = 7, 2*3 + 5 = 11, 2*2 + 1 = 5, 2*1 + 3 = 5, 2*3 + 1 = 7, 2*1 + 5 = 7 are all prime.
a(10) = 1 since 10 = 3 + 4 + 3, and 2*3 + 1 = 7, 2*4 + 3 = 11, 2*3 + 5 = 11 are all prime.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
TQ[n_]:=SQ[8n+1]
a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[2j+3]&&PrimeQ[2(n-i-j)+5]&&TQ[i*j(n-i-j)], 1, 0], {i, 1, n-2}, {j, 1, n-1-i}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 19 2013
STATUS
approved