If the number of 1's is divisible by 3, then it is divisible by 37. If the number of 1's is 1 mod 3, then the number of 1's is either 1 or 4 mod 6. If it is 1 mod 6, then the number is divisible by 7. If it is 4 mod 6, then the number is divisible by 13. If the number of 1's is 2 mod 3, then it is divisible by 3. Therefore, no number in the sequence 371, 3711, 37111... is prime.

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