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.

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

Offset

1,8

Comments

Conjecture: a(n) > 0 if n is not a divisor of 12.

Clearly, this implies the twin prime conjecture.

Links

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

Crossrefs

Cf. A000040, A001359, A006512, A236531, A236831.

Sequence in context: A319081 A336931 A363953 * A308778 A127284 A120691

Adjacent sequences: A182659 A182660 A182661 * A182663 A182664 A182665

Keyword

nonn

Author

Zhi-Wei Sun, Jan 31 2014

Status

approved