A236531 a(n) = |{0 < k < n: {6*k -1 , 6*k + 1} and {prime(n-k), prime(n-k) + 2} are both twin prime pairs}|.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 27 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 3 is neither 11 nor 125, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1, prime(m) + 2 and 3*prime(m) - 10 are all prime.

(iii) Any integer n > 458 can be written as p + q with q > 0 such that {p, p + 2} and {prime(q), prime(q) + 2} are both twin prime pairs.

This is much stronger than the twin prime conjecture. We have verified part (i) of the conjecture for n up to 2*10^7.

Examples

			a(11) = 1 since {6*1 - 1, 6*1 + 1} = {5, 7} and {prime(10), prime(10) + 2} = {29, 31} are both twin prime pairs.
a(16) = 1 since {6*3 - 1, 6*3 + 1} = {17, 19} and {prime(13), prime(13) + 2} = {41, 43} are both twin prime pairs.
		

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

Crossrefs

Cf. A000040, A001359, A002822, A006512, A199920.

Programs

Mathematica

p[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
q[n_]:=PrimeQ[Prime[n]+2]
a[n_]:=Sum[If[p[k]&&q[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]