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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 (list; graph; refs; listen; history; text; internal format)
 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 Zhi-Wei Sun, Table of n, a(n) for n = 1..6000 Zhi-Wei Sun, A new conjecture on triangular numbers, a message to Number Theory List, Oct. 25, 2013. 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}] CROSSREFS Cf. A000040, A000217, A068307, A132399, A227877, A229166, A230121, A230141, A230219, A230596. Sequence in context: A208384 A327800 A286236 * A286239 A341585 A343866 Adjacent sequences:  A230448 A230449 A230450 * A230452 A230453 A230454 KEYWORD nonn AUTHOR Zhi-Wei Sun, Oct 19 2013 STATUS approved

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Last modified July 23 21:10 EDT 2021. Contains 346265 sequences. (Running on oeis4.)