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I was wondering if a Smith number can be a Lucas Carmichael number.

Is there a proof that there is no such number?

If there is such a number, can you tell me it and how you got it?

I have written a brute force program and so far I have found no such number between 1 and 1,000,000.

Thanks, Remy

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    $\begingroup$ Please define both kinds of numbers in your question. $\endgroup$
    – D. Ror.
    Oct 20, 2016 at 16:54
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    $\begingroup$ I guess en.wikipedia.org/wiki/Smith_number and en.wikipedia.org/wiki/Lucas-Carmichael_number $\endgroup$ Oct 20, 2016 at 16:59
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    $\begingroup$ I suppose one could argue that this question is rather unmotivated, since it takes two "random" categories of numbers, very loosely related, and asks for their intersection. Of all possible categories of numbers, do you have any reason to choose these two particular ones? $\endgroup$
    – Wojowu
    Oct 20, 2016 at 17:26
  • $\begingroup$ Upvoted because I want to see the answer, but I agree, it is very unmotivated. $\endgroup$ Oct 20, 2016 at 20:04
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    $\begingroup$ “Smith numbers were named by Albert Wilansky ... He noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith.” Mind boggles. $\endgroup$ Oct 20, 2016 at 20:15

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Here are all numbers below $10^9$ that are both Smith and Lucas-Carmichael:

8164079, 8421335, 21408695, 30071327, 47324639, 77350559, 103727519, 121538879, 134151479, 202767551, 239875559, 287432495, 306871487, 466861199, 560974259, 566019167, 574342145, 592557119, 594633599, 602758079, 677913599, 832477799

UPDATE. These numbers now form http://oeis.org/A277580

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    $\begingroup$ Hey – there's my phone number! $\endgroup$ Oct 21, 2016 at 5:20
  • $\begingroup$ @GerryMyerson: Now you know its purpose. $\endgroup$ Oct 21, 2016 at 14:27
  • $\begingroup$ Thank You, if I may, how did you get these numbers? Is there some formula, algebra, or did you brute force check? Thank you again $\endgroup$ Oct 22, 2016 at 19:28
  • $\begingroup$ @RemyRouyer: Just a kind of brute force. $\endgroup$ Oct 23, 2016 at 18:33

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