

A230140


Number of ways to write n = x + y + z with 0 < x <= y <= z such that 6*x1, 6*y1, 6*z1 are among those primes p (terms of A230138) with p + 2 and 2*p  5 also prime.


9



0, 0, 1, 1, 2, 2, 3, 2, 3, 1, 2, 2, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 4, 2, 5, 2, 6, 3, 6, 5, 4, 5, 3, 5, 5, 8, 7, 6, 5, 6, 5, 5, 7, 6, 8, 4, 6, 5, 6, 7, 9, 8, 8, 5, 7, 6, 8, 10, 6, 10, 4, 8, 6, 6, 10, 6, 9, 5, 6, 5, 7, 7, 9, 6, 7, 8, 5, 10, 6, 9, 6, 6, 7, 4, 7, 7, 9, 6, 5, 5, 4, 6, 5, 6, 5, 5, 6, 4, 6, 6
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OFFSET

1,5


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 2, i.e., 6*n3 with n > 2 can be expressed as a sum of three terms from A230138. Moreover, for any integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x1, 6*y1, 6*z1 are primes in A230138.
(ii) For each integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x1, 6*y1, 6*z1 are among those primes p with p + 2 and 2*p + 9 also prime.
Note that part (i) of this conjecture implies that there are infinitely many primes in A230138.
Indices k such that a(m)>a(k) for all m>k, are (2, 10, 26, 334, 439, 544, 551, 684, ...). The only sequence which has the first 5 terms within the 3 lines of data is A212067. (Certainly a coincidence.)  M. F. Hasler, Oct 10 2013


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) = 1 since 10 = 2 + 3 + 5, and the three numbers 6*21=11, 6*31=17 and 6*51=29 are terms of A230138.


MATHEMATICA

SQ[n_]:=PrimeQ[6n1]&&PrimeQ[6n+1]&&PrimeQ[12n7]
a[n_]:=Sum[If[SQ[i]&&SQ[j]&&SQ[nij], 1, 0], {i, 1, n/3}, {j, i, (ni)/2}]
Table[a[n], {n, 1, 100}]


PROG

(PARI) ip(x)=isprime(6*x1) && isprime(6*x+1) && isprime(12*x7); a(n)=sum(x=1, n\3, sum(y=x, ip(x)*(nx)\2, ip(y) && ip(nxy))) \\  M. F. Hasler, Oct 10 2013


CROSSREFS

Cf. A068307, A230138, A230141.
Sequence in context: A279632 A300817 A341417 * A156220 A261653 A347658
Adjacent sequences: A230137 A230138 A230139 * A230141 A230142 A230143


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 10 2013


STATUS

approved



