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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 ?

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  • $\begingroup$ Note that the sequence ends at $37\cdot27=999$ (the next one would be $37\cdot30 = 1110)$. $\endgroup$ – quasi Jul 13 '17 at 4:51
  • $\begingroup$ Note that after $111$, the rest are unremarkable, because what you have is $37\cdot3, 37\cdot3\cdot2, 37\cdot3\cdot3$, and so on. I would start by looking at the prime factorisations of $1111, 11111 $ and so on. By the way, does $11$ count as such a prime? $\endgroup$ – Arthur Jul 13 '17 at 4:55
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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.

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