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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.
3
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
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 05 2014
STATUS
approved