A034304: Nonprime; becomes prime if any digit is deleted (zeros not allowed in the number). 

Sounds crunchy and active in the first 3 seconds, then there are large pockets of inactivity till the end.

Residue counts:
2   [4, 169]
3   [60, 18, 95]
4   [3, 95, 1, 74]
5   [4, 49, 64, 35, 21]
6   [1, 16, 1, 59, 2, 94]
7   [24, 24, 26, 23, 21, 30, 25]
8   [2, 44, 0, 35, 1, 51, 1, 39]
9   [25, 8, 28, 20, 3, 39, 15, 7, 28]
10   [0, 49, 4, 35, 0, 4, 0, 60, 0, 21]
11   [43, 16, 10, 12, 7, 14, 26, 9, 17, 6, 13]
12   [1, 10, 0, 26, 1, 52, 0, 6, 1, 33, 1, 42]

First 100 first differences:
[3, 2, 5, 1, 2, 17, 3, 2, 15, 3, 2, 34, 6, 2, 52, 200, 40, 2, 4, 20, 
34, 2, 138, 60, 40, 2, 18, 648, 18, 276, 66, 198, 54, 242, 44, 294, 6, 
234, 302, 18, 42, 138, 6, 162, 720, 132, 102, 6, 60, 132, 360, 586, 300, 
554, 342, 846, 114, 478, 28, 38, 300, 234, 66, 314, 720, 60, 6, 1894, 
906, 2094, 62, 60, 6, 1638, 360, 880, 1856, 4378, 3300, 4226, 3054, 
2226, 5194, 5882, 1344, 396, 2600, 9400, 264, 9880, 2640, 300, 60, 
540, 60, 5640, 54, 1522, 11440, 204]

The sequence is extremely sparse with 173 terms under 10^6. One interesting 
observation is that there are only 4 even terms in the sequence. Notice that 
if an even term had three or more digits, then deleting any digit that is not 
the last digit would leave a number that is still even and greater than 2, so 
it could not be prime. Therefore, any even term in this sequence can only have 
two digits. Since deleting the first digit must give a prime number, the last 
digit has to be 2, the only even prime. Deleting the last digit must also give 
a prime number, so the first digit must be one of 2, 3, 5, or 7. Thus, the only 
possible even terms are 22, 32, 52, and 72.

-- Lauren Yee