A247946 Primes in the tetranacci sequence A000288.
7, 13, 181, 349, 673, 1297, 34513, 90799453, 175021573, 4657290577, 17304140641, 1131469145856472270556751793, 1544310310927991136025089626209, 1442398599584422734286432395814518441223501, 18598135820391234761502881488353916158281807617671450769
Offset: 1
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Mathematica
a={1,1,1,1}; For[n=4, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[4]]=sum] Select[LinearRecurrence[{1,1,1,1},{1,1,1,1},300],PrimeQ] (* Harvey P. Dale, Jan 15 2015 *)
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