
COMMENTS

The next term may or may not be 3719.
Comment from Hugo Pfoertner, May 28 2019: At the moment, the sequence is confirmed for all terms < 3719. So far, it is not known whether (3719*4^k+1)/3 is composite for all k > 50000.
Updates from Hugo Pfoertner, Jun 05 2019 and Jun 24 2019: All numbers marked with ? in the list below are checked through k <= 200000. No other numbers with unknown status exist below 25000.
The sequence may continue as follows: 3719?, 4889, 5459?, 5561, 5771, 6341, 6509, 7271, 7829, 8609, 9959, 10019?, 10289?, 10799, 10841, 11171, 11429, 12809?, 13079, 13751, 13961, 14279?, 14531, 14699, 15419?, 15461, 16019, 16799, 18149, 18971?, 18989, 19031, 19361, 20261?, 21269, 21941, 22151, 22529?, 22721, 22889, 23099?, 23651, 24209, 24989


EXAMPLE

419 is in the sequence. Consider (419*4^m+1)/3: for m == 0 (mod 6) it is divisible by 5 and 7, for m == 1 (mod 6) divisible by 13, for m == 2 (mod 6) divisible by 3 and 5, for m == 3 (mod 6) divisible by 7, for m == 4 (mod 6) divisible by 5 and for m == 5 (mod 6) divisible by 3. The "mod 6" is derived from the fact that 4^61 = 3^2*5*7*13, so 4^6 == 1 (mod 5), 4^6 == 1 (mod 7), 4^6 = 1 (mod 9) and 4^6 == 1 (mod 13).  Andrew Weimholt, May 26 2019
6971 is not in the sequence, because (6971*4^204688+1)/3 is a prime with 123238 decimal digits.  Hugo Pfoertner, Jun 05 2019
