

A229969


Number of ways to write n = x + y + z with 0 < x <= y <= z such that all the six numbers 2*x1, 2*y1, 2*z1, 2*x*y1, 2*x*z1, 2*y*z1 are prime.


5



0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 4, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 2, 7, 4, 3, 5, 3, 2, 6, 3, 4, 3, 4, 5, 3, 4, 6, 6, 3, 5, 4, 5, 6, 9, 4, 8, 4, 7, 10, 2, 6, 12, 9, 1, 7, 7, 6, 12, 10, 3, 7, 8, 8, 9, 9, 5, 3, 7, 3, 7, 3, 9, 10, 8, 6, 11, 11, 13, 15, 6, 6, 10, 15, 11, 11, 13, 8, 12, 12, 7, 10, 8, 13, 12
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OFFSET

1,10


COMMENTS

Conjecture: a(n) > 0 for all n > 5. Moreover, any integer n > 6 can be written as x + y + z with x among 3, 4, 6, 10, 15 such that 2*y1, 2*z1, 2*x*y1, 2*x*z1, 2*y*z1 are prime.
We have verified this conjecture for n up to 10^6. As (2*x1)+(2*y1)+(2*z1) = 2*(x+y+z)3, it implies Goldbach's weak conjecture which has been proved.
ZhiWei Sun also had some similar conjectures including the following (i)(iii):
(i) Any integer n > 6 can be written as x + y + z (x, y, z > 0) with 2*x1, 2*y1, 2*z1 and 2*x*y*z1 all prime and x among 2, 3, 4. Also, each integer n > 2 can be written as x + y + z (x, y, z > 0) with 2*x+1, 2*y+1, 2*z+1 and 2*x*y*z+1 all prime and x among 1, 2, 3.
(ii) Each integer n > 4 can be written as x + y + z with x = 3 or 6 such that 2*y+1, 2*x*y*z1 and 2*x*y*z+1 are prime.
(iii) Every integer n > 5 can be written as x + y + z (x, y, z > 0) with x*y1, x*z1, y*z1 all prime and x among 2, 6, 10. Also, any integer n > 2 not equal to 16 can be written as x + y + z (x, y, z > 0) with x*y+1, x*z+1, y*z+1 all prime and x among 1, 2, 6.
See also A229974 for a similar conjecture involving three pairs of twin primes.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.


EXAMPLE

a(10) = 2 since 10 = 2+2+6 = 3+3+4 with 2*21, 2*61, 2*2*21, 2*2*6 1, 2*31, 2*41, 2*3*31, 2*3*41 all prime.


MATHEMATICA

a[n_]:=Sum[If[PrimeQ[2i1]&&PrimeQ[2j1]&&PrimeQ[2(nij)1]&&PrimeQ[2i*j1]&&PrimeQ[2i(nij)1]&&PrimeQ[2j(nij)1], 1, 0], {i, 1, n/3}, {j, i, (ni)/2}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A068307, A219842, A219864, A227923, A229974.
Sequence in context: A186728 A158298 A009191 * A260909 A114717 A318670
Adjacent sequences: A229966 A229967 A229968 * A229970 A229971 A229972


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 04 2013


STATUS

approved



