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

1,3

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.

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(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.

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}]

Crossrefs

Cf. A000040, A001359, A006512, A218829, A237260.

Sequence in context: A369705 A134868 A322861 * A235614 A127417 A308622

Adjacent sequences: A237256 A237257 A237258 * A237260 A237261 A237262

Keyword

nonn

Author

Zhi-Wei Sun, Feb 05 2014

Status

approved