A234703 Primes in the tribonacci-like sequence, A214727.
2, 2, 5, 101, 13241, 151537, 66848890001808737, 8602289657912317933269334679427588251509673524841616601
Offset: 1
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Mathematica
a={1,2,2}; Print[2]; Print[2]; For[n=3, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[3]]=sum] Select[LinearRecurrence[{1,1,1},{1,2,2},500],PrimeQ] (* Harvey P. Dale, Feb 22 2023 *)
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