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Let f(n)=$1234567891011$...n (i.e. concatenating the first n integers). And consider this sequence of numbers n:

  • 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....

So we have this sequence : $2,3,25,43,157,19,271,229,49,31,181,91,397,481,301,127,511,439,2701,37,79,253,793,1531,1237,523,1867,637$. My question : What is the smallest value of n such that $619$ is the least prime factor of f(n)?

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I used a simple Perl script with Math::GMPz, same as for Kenan's almost identical question a few days ago. I noted there that 619 and 947 both had entries > 50k.

n=64033 for prime 619.

n=145591 for prime 947.

n=176053 for prime 1613, which is the largest n for primes under 2000.

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