A235873 Primes in the tribonacci-like sequence, A141523.
3, 5, 7, 13, 83, 281, 3217, 10883, 1425427, 55187617, 24453221203, 124001884480009, 29872617402415741, 185875267730565697, 341877918058715653, 44580781450601596678810171573, 36012536557658790037420884825332617431175065740791
Offset: 1
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Mathematica
a={3,1,1}; Print[3]; For[n=3, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[3]]=sum]