

A182662


Number of ordered ways to write n = p + q with q > 0 such that p, 3*(p + prime(q))  1 and 3*(p + prime(q)) + 1 are all prime.


5



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

1,8


COMMENTS

Conjecture: a(n) > 0 if n is not a divisor of 12.
Clearly, this implies the twin prime conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(11) = 1 since 11 = 7 + 4 with 7, 3*(7 + prime(4))  1 = 3*14  1 = 41 and 3*(7 + prime(4)) + 1 = 3*14 + 1 = 43 all prime.
a(210) = 1 since 210 = 97 + 113 with 97, 3*(97 + prime(113))  1 = 3*(97 + 617)  1 = 2141 and 3*(97 + prime(113)) + 1 = 3*(97 + 617) + 1 = 2143 all prime.


MATHEMATICA

p[n_, m_]:=PrimeQ[3(m+Prime[nm])1]&&PrimeQ[3(m+Prime[nm])+1]
a[n_]:=Sum[If[p[n, Prime[k]], 1, 0], {k, 1, PrimePi[n1]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A001359, A006512, A236531, A236831.
Sequence in context: A089339 A249303 A319081 * A308778 A127284 A120691
Adjacent sequences: A182659 A182660 A182661 * A182663 A182664 A182665


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 31 2014


STATUS

approved



