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There's a post in CodeGolf which asks for code to find numbers whose digits contain their prime factors without rearrangement. The author suggests the mathematical definition is

"Determine if the number n is a composite number such that all prime factors of n are a subsequence of n"

What I'm wondering is: are there any such numbers where not only are all prime factors found, but, if a prime factor is repeated N times in the factorization, then that prime number can be found in N independent subsequences within the composite number.

Or even more difficult: if we only allow contiguous subsequences, do any such numbers exist?

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  • $\begingroup$ Any prime number where $1$ appears as a digit exactly once will work. $\endgroup$
    – user44191
    Commented Jan 26, 2021 at 19:15
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    $\begingroup$ @user44191 sorry, I should have emphasized the composite number part of the conjectures $\endgroup$
    – Carl Witthoft
    Commented Jan 26, 2021 at 21:11
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    $\begingroup$ Thinking about it a bit, I believe that a composite number can have at most the same number of digits as its prime factors combined, so the multiset of the digits of the number would have to be equal to the multiset of the digits of all the prime factors. i.e. you can't have a bigger or smaller number of digits in the number than you have in the combined prime factors. Don't know if that helps at all. $\endgroup$
    – IronEagle
    Commented Jan 28, 2021 at 17:33
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    $\begingroup$ If you're interested in the substring, rather than subsequence, version where repeats are allowed, it's oeis.org/A050694 $\endgroup$
    – IronEagle
    Commented Jan 29, 2021 at 3:45
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    $\begingroup$ Checking up to 2^31 still reveals no solutions. $\endgroup$
    – IronEagle
    Commented Sep 18, 2022 at 15:58

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