A074311 a(1) = 1; a(2) = 2; a(n) = smallest number greater than the previous term such that the average of three successive terms is a prime.

1, 2, 3, 4, 8, 9, 16, 26, 27, 34, 50, 57, 70, 74, 75, 88, 104, 111, 112, 116, 153, 178, 188, 207, 238, 242, 243, 268, 278, 285, 286, 308, 327, 358, 362, 381, 394, 416, 417, 424, 452, 453, 466, 470, 501, 502, 506, 519, 538, 566, 567, 574, 590, 597, 610, 614, 615