A073670 a(1) = 1, a(n) = the smallest composite multiple of n not included earlier such that every partial sum is prime.

1, 4, 6, 8, 10, 12, 42, 24, 72, 20, 132, 36, 52, 84, 60, 80, 34, 144, 38, 160, 168, 44, 46, 96, 50, 130, 54, 56, 58, 90, 62, 256, 264, 306, 210, 180, 74, 228, 78, 240, 492, 126, 774, 88, 450, 230, 282, 192, 294, 100, 209, 109, 321, 113, 883, 127, 355, 131, 485