A237259 Least positive integer k < n such that prime(k) + 2 and prime(prime(n-k)) + 2 are both prime, or 0 if such a number k does not exist.

0, 0, 2, 2, 2, 2, 3, 2, 2, 3, 5, 5, 7, 7, 2, 2, 3, 5, 5, 7, 7, 20, 10, 10, 2, 2, 3, 5, 5, 7, 2, 2, 3, 5, 5, 7, 7, 35, 10, 10, 17, 2, 3, 20, 5, 17, 7, 35, 20, 10, 28, 28, 13, 41, 26, 26, 17, 28, 35, 20, 2, 3, 33, 5, 2, 3, 2, 3, 10, 5

Offset: 1

Author

Zhi-Wei Sun, Feb 05 2014

Keywords

• nonn

Comments

Conjecture: a(n) < 2*sqrt(n)*log(3*n) for all n > 0.

We have verified this for n up to 5*10^5. Note that a(202) = 173 > 2*sqrt(202)*log(2*202).

According to the conjecture in A218829, a(n) should be positive for all n > 2.

Examples

			a(3) = 2 since prime(2) + 2 = 5 and prime(prime(3-2)) + 2 = prime(2) + 2 = 5 are both prime, but prime(1) + 2 = 4 is composite.
		

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, A006512, A218829, A237260.

Programs

Mathematica

pq[k_,m_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[m]]+2]
Do[Do[If[pq[k,n-k],Print[n," ",k];Goto[aa]],{k,1,n-1}];
Print[n," ",0];Label[aa];Continue,{n,1,70}]