A136851 a(1)=3 (smallest odd prime); for n>=2, a(n) = smallest prime such that 1) a(n)-a(n-1) = prime if odd, else double of prime and 2) for sum_{ k = 1..n } a(k) = prime if odd, else double of prime.

3, 7, 13, 23, 37, 59, 97, 107, 241, 251, 613, 911, 937, 983, 997, 1223, 1381, 1499, 1621, 1979, 2281, 2399, 2521, 2663, 2677, 3299, 3361, 3623, 3637, 3671, 3697, 3803, 3889, 3911, 4273, 4451, 4729, 4787, 4909, 5003, 5821, 5927, 6229, 6791, 6793, 6827, 6829