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 A230037 Number of ways to write n = x + y + z (0 < x <= y <= z) such that the four pairs {6*x-1, 6*x+1}, {6*y-1, 6*y+1}, {6*z-1, 6*z+1} and {6*x*y-1, 6*x*y+1} are twin prime pairs. 4
 0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 4, 3, 2, 3, 2, 5, 2, 4, 3, 4, 4, 4, 3, 3, 4, 5, 7, 4, 5, 2, 5, 4, 5, 7, 5, 5, 4, 4, 4, 6, 6, 8, 4, 5, 3, 4, 5, 6, 7, 4, 6, 2, 5, 3, 7, 8, 4, 4, 1, 4, 2, 7, 6, 3, 5, 3, 5, 4, 6, 6, 5, 4, 3, 5, 4, 5, 3, 3, 3, 6, 7, 5, 2, 4, 4, 5, 3, 6, 4, 3, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n) > 0 for all n > 2. Moreover, any integer n > 2 can be written as x + y + z with x = 1 or 5 such that {6*y-1, 6*y+1}, {6*z-1, 6*z+1} and {6*x*y-1, 6*x*y+1} are twin prime pairs. We have verified this for n up to 5*10^7. It implies the twin prime conjecture. Zhi-Wei Sun also made the following similar conjectures: (i) Any integer n > 2 can be written as x + y + z (x, y, z > 0) with the 8 numbers 6*x-1, 6*x+1, 6*y-1, 6*y+1, 6*z-1, 6*z+1, 6*x*y-1 and 6*x*y*z-1 (or 12*x*y-1) all prime. (ii) Each integer n > 2 can be written as x + y + z (x, y, z > 0) with the 8 numbers 6*x-1, 6*x+1, 6*y-1, 12*y-1, 6*z-1 (or 6*x*y-1), 2*(x^2+y^2)+1, 2*(x^2+z^2)+1, 2*(y^2+z^2)+1 all prime. (iii) Any integer n > 8 can be written as x + y + z (x, y, z > 0) with x-1, x+1, y-1, y+1, x*z-1 and y*z-1 all prime. (iv) Every integer n > 4 can be written as p + q + r (r > 0) with p, q, 2*p*q-1, 2*p*r-1 and 2*q*r-1 all prime. (v) Any integer n > 10 can be written as x^2 + y^2 + z (x, y, z > 0) with 2*x*y-1, 2*x*z+1 and 2*y*z+1 all prime. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Two conjectures involving six primes, a message to Number Theory List, Oct. 5, 2013. Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588. EXAMPLE a(10) = 1 since 10 = 1 + 2 + 7 , and {6*1-1, 6*1+1}, {6*2-1, 6*2+1}, {6*7-1, 6*7+1}  and {6*1*2-1, 6*1*2+1} are twin prime pairs. MATHEMATICA a[n_]:=Sum[If[PrimeQ[6i-1]&&PrimeQ[6i+1]&&PrimeQ[6j-1]&&PrimeQ[6j+1]&&PrimeQ[6i*j-1] &&PrimeQ[6*i*j+1]&&PrimeQ[6(n-i-j)-1]&&PrimeQ[6(n-i-j)+1], 1, 0], {i, 1, n/3}, {j, i, (n-i)/2}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A001359, A006512, A219842, A219864, A229969, A229974, A230040. Sequence in context: A268173 A008617 A025824 * A211262 A185319 A161232 Adjacent sequences:  A230034 A230035 A230036 * A230038 A230039 A230040 KEYWORD nonn AUTHOR Zhi-Wei Sun, Oct 06 2013 STATUS approved

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)