

A229974


Number of ways to write n = x + y + z (x, y, z > 0) with the six numbers 2*x1, 2*x+1, 2*x*y1, 2*x*y+1, 2*x*y*z1, 2*x*y*z+1 all prime.


5



0, 0, 0, 1, 1, 4, 2, 1, 2, 4, 5, 3, 3, 8, 1, 9, 4, 6, 3, 8, 16, 8, 4, 8, 7, 3, 10, 7, 3, 14, 4, 6, 8, 13, 12, 14, 6, 8, 13, 7, 13, 15, 13, 9, 9, 10, 7, 13, 14, 7, 16, 15, 12, 8, 16, 31, 11, 6, 16, 13, 16, 15, 26, 8, 10, 17, 10, 12, 11, 17, 9, 9, 13, 18, 17, 23, 14, 10, 7, 13, 29, 13, 18, 14, 9, 19, 21, 14, 19, 14, 25, 11, 14, 18, 13, 21, 15, 26, 14, 8
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OFFSET

1,6


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 3. Moreover, any integer n > 3 can be written as x + y + z with x among 2, 3, 6 such that {2*x*y1, 2*x*y+1} and {2*x*y*z1, 2*x*y*z+1} are twin prime pairs.
(ii) Each integer n > 11 can be written as x + y + z (x, y, z > 0) with x1, x+1, x*y1, x*y+1, x*y*z1, x*y*z+1 all prime, moreover we may require that x is among 4, 6, 12.
(iii) Any integer n > 3 not equal to 10 can be written as x + y + z (x, y, z > 0) such that the three numbers 2*x1, 2*x*y1 and 2*x*y*z1 are Sophie Germain primes, moreover we may require that x is among 2, 3, 6.
Note that part (i) or (ii) of the above conjecture implies the twin prime conjecture, while part (iii) implies that there are infinitely many Sophie Germain primes.
See also the comments of A229969 for other similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..4000
ZhiWei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.


EXAMPLE

a(4) = 1 since 4 = 2+1+1 with 2*21 and 2*2+1 both prime.
a(5) = 1 since 5 = 3+1+1 with 2*31 and 2*3+1 both prime.
a(15) = 1 since 15 = 6+5+4 with 2*61, 2*6+1, 2*6*51, 2*6*5+1, 2*6*5*41, 2*6*5*4+1 all prime.


MATHEMATICA

a[n_]:=Sum[If[PrimeQ[2i1]&&PrimeQ[2i+1]&&PrimeQ[2*i*j1]&&PrimeQ[2i*j+1]&&PrimeQ[2i*j*(nij)1]&&PrimeQ[2i*j*(nij)+1], 1, 0], {i, 1, n2}, {j, 1, n1i}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A001359, A006512, A005384, A229969, A219842, A219864.
Sequence in context: A270047 A132116 A327252 * A281065 A280988 A175665
Adjacent sequences: A229971 A229972 A229973 * A229975 A229976 A229977


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 05 2013


STATUS

approved



