Sorry, I can't post comments yet (maybe never:))!

I wrote a program to find such numbers in the intervals [2,4000), [10<sup>6</sup>,10<sup>6</sup>+4000), [10<sup>9</sup>,10<sup>9</sup>+4000), [10<sup>12</sup>,10<sup>12</sup>+4000), and [10<sup>15</sup>,10<sup>15</sup>+4000)

Here are the numbers and signatures:
https://pastebin.com/piMZNQKx