Is $ 37$ the only prime satisfying the below property?

Question

Really i have tried to know more about behavior of primes to produce some sequences i have got that $37$ satisfy the below property :

Property:

$37*3=111,37*6=222,37*9=333,\cdots $ , Now I ask if $ 37$ is the only prime satisfying that property to get a sequence which it include numbers ,each number have same digit as shown before ?

Answer 1

How about $3?$ $3\cdot 37=111, 3\cdot 74=222,$ etc.

Another is $271: 271 \cdot 41=11111, 271\cdot 82=22222,$ etc.

Take any repunit, factor it, and choose one of the primes.

I suspect you have something more restrictive in mind, but this seems to meet the request.