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All terms are == 1 mod 4, hence in all cases p+3 is divisible by 4 (and is not squarefree).
Table of n, a(n) for n=1..51.
Select[Prime[Range], SquareFreeQ[# + 1] && SquareFreeQ[# + 2] &]
(MAGMA) [p: p in PrimesUpTo(1500) | IsSquarefree(p+1) and IsSquarefree(p+2)]; // Vincenzo Librandi, Feb 09 2016
Intersection of A049097 and A049233.
Sequence in context: A224339 A133204 A207040 * A152658 A100877 A261580
Adjacent sequences: A268611 A268612 A268613 * A268615 A268616 A268617
Zak Seidov, Feb 08 2016