

A230037


Number of ways to write n = x + y + z (0 < x <= y <= z) such that the four pairs {6*x1, 6*x+1}, {6*y1, 6*y+1}, {6*z1, 6*z+1} and {6*x*y1, 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
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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*y1, 6*y+1}, {6*z1, 6*z+1} and {6*x*y1, 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.
ZhiWei 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*x1, 6*x+1, 6*y1, 6*y+1, 6*z1, 6*z+1, 6*x*y1 and 6*x*y*z1 (or 12*x*y1) all prime.
(ii) Each integer n > 2 can be written as x + y + z (x, y, z > 0) with the 8 numbers 6*x1, 6*x+1, 6*y1, 12*y1, 6*z1 (or 6*x*y1), 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 x1, x+1, y1, y+1, x*z1 and y*z1 all prime.
(iv) Every integer n > 4 can be written as p + q + r (r > 0) with p, q, 2*p*q1, 2*p*r1 and 2*q*r1 all prime.
(v) Any integer n > 10 can be written as x^2 + y^2 + z (x, y, z > 0) with 2*x*y1, 2*x*z+1 and 2*y*z+1 all prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Two conjectures involving six primes, a message to Number Theory List, Oct. 5, 2013.
ZhiWei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.


EXAMPLE

a(10) = 1 since 10 = 1 + 2 + 7 , and {6*11, 6*1+1}, {6*21, 6*2+1}, {6*71, 6*7+1} and {6*1*21, 6*1*2+1} are twin prime pairs.


MATHEMATICA

a[n_]:=Sum[If[PrimeQ[6i1]&&PrimeQ[6i+1]&&PrimeQ[6j1]&&PrimeQ[6j+1]&&PrimeQ[6i*j1]
&&PrimeQ[6*i*j+1]&&PrimeQ[6(nij)1]&&PrimeQ[6(nij)+1], 1, 0], {i, 1, n/3}, {j, i, (ni)/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

ZhiWei Sun, Oct 06 2013


STATUS

approved



