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a(1) = 5; for n > 1, a(n) = prime p >= a(n-1) such that both q = p + 2n and r = q + 2n + 2 are primes.
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%I #9 Mar 02 2017 13:15:10

%S 5,7,17,23,31,41,53,67,71,89,127,149,173,199,251,251,277,347,383,409,

%T 461,479,523,593,641,691,719,773,823,887,971,1033,1097,1163,1231,1301,

%U 1373,1447,1619,1709,1741,1823,1907,1951,1979,2087,2143,2243,2243

%N a(1) = 5; for n > 1, a(n) = prime p >= a(n-1) such that both q = p + 2n and r = q + 2n + 2 are primes.

%t m = 0; p = 3; s = {}; Do[m = m + 2; While[! PrimeQ[p + m] || ! PrimeQ[p + 2*m + 2], p = NextPrime[p]]; AppendTo[s, p], {50}]; s

%Y Cf. A283145.

%K nonn

%O 1,1

%A _Zak Seidov_, Mar 01 2017