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1 solution
As the sequence seemed to have random and unpredictable intervals I thought a good thing to try was to see if it had some link with prime numbers
the first 35 primes are : 2 , 3 , 5 , 7 , 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 2 9 , 3 1 , 3 7 , 4 1 , 4 3 , 4 7 , 5 3 , 5 9 , 6 1 , 6 7 , 7 1 , 7 3 , 7 9 , 8 3 , 8 9 , 9 7 , 1 0 1 , 1 0 3 , 1 0 7 , 1 0 9 , 1 1 3 , 1 2 7 , 1 3 1 , 1 3 7 , 1 3 9 , 1 4 9
I noticed that the numbers in the sequence came between some of these primes 4 i s b e t w e e n 3 a n d 5 6 i s b e t w e e n 5 a n d 7 1 2 i s b e t w e e n 1 1 a n d 1 3 . . . a n d s o o n From that you can see that the numbers in the sequence are the numbers in between the primes that differ by only 2. So the sequence is made up of the numbers between the pairs of twin primes. The next set of twin primes following 107 and 109 are 137 and 139 so the next number in the sequence is 1 3 8