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Let f(n)=$1234567891011$....n

(concatenation of first n natural numbers). I make a sequence of numbers made with this following definition: Smallest number n such that the m-th prime number is the least prime factor/divisor of f(n). And I found these following:

  • n=2 is the first case where 2 is the least prime factor of f(n)
  • n=3 is the first case where 3 is the least prime factor of f(n)
  • n=25 is the first case where 5 is the least prime factor of f(n)

  • n=43 is the first case where 7 is the least prime factor of f(n)

  • n=157 is the first case where 11 is the least prime factor of f(n)

  • n=19 is the first case where 13 is the least prime factor of f(n)

  • n=271 is the first case where 17 is the least prime factor of f(n)

  • And so on...

  • And quite far: n=793 is the first case where $83$ is the least prime factor of f(n)

My question: When is the first case where 67 is the least prime factor of f(n)? (Can 67 ever be the least prime factor of f(n)?). Additional question: Is there a prime number that will NEVER be the least prime divisor of f(n) ?

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  • $\begingroup$ Here are all solutions below $10^4.$ $\endgroup$
    – Lucian
    Commented Dec 9, 2015 at 20:15
  • $\begingroup$ @Lucian, you mean all of them have least prime factor of 67 ? (thanks for that information!) : D $\endgroup$
    – Kenan Guon
    Commented Dec 9, 2015 at 21:24

1 Answer 1

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I think it is 2701. Checking in some more details.

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  • $\begingroup$ you mean n=2701 is the first case where 67 is the least prime factor of f(n)? $\endgroup$
    – Kenan Guon
    Commented Dec 9, 2015 at 17:07
  • $\begingroup$ Yes, that's what you were looking for, right? I wrote a program in Java and it says 2701 is the number. $\endgroup$ Commented Dec 9, 2015 at 17:08
  • $\begingroup$ What about least prime factor 89 and 97 ? ( I found n=523 for least prime factor 101) : D $\endgroup$
    – Kenan Guon
    Commented Dec 9, 2015 at 17:10
  • $\begingroup$ It's 1531 for 89. $\endgroup$ Commented Dec 9, 2015 at 17:17
  • $\begingroup$ It's 1237 for 97. $\endgroup$ Commented Dec 9, 2015 at 17:23

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