It is easy to show that 37 followed by n "ones" is always a composite, since  the resulting number  is divisible by at least one of the following primes: 3, 7, 13, 37:

If n= 0 mod3  then it is divisible  by 37      since 111=3*37

If n= 2 mod3  then it is divisible  by 3   since the sod is a multiple of 3

If n= 1 mod6  then it is divisible  by 7   since 111111=3*37*7*11*13 and 371=7*53

If n= 4 mod6  then it is divisible  by 13  since111111=3*37*7*11*13

and 371111=260000+111111

Bonus finding:   the same applies to 38  in lieu of 37:   381, 3811, 38111…Prove it.