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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 '16 at 16:54
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    $\begingroup$ I guess en.wikipedia.org/wiki/Smith_number and en.wikipedia.org/wiki/Lucas-Carmichael_number $\endgroup$ – Pietro Majer Oct 20 '16 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 '16 at 17:26
  • $\begingroup$ Upvoted because I want to see the answer, but I agree, it is very unmotivated. $\endgroup$ – Juan Sebastian Lozano Oct 20 '16 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. $\endgroup$ – Emil Jeřábek Oct 20 '16 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! $\endgroup$ – Gerry Myerson Oct 21 '16 at 5:20
  • $\begingroup$ @GerryMyerson: Now you know its purpose. $\endgroup$ – Max Alekseyev Oct 21 '16 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 $\endgroup$ – Remy Rouyer Oct 22 '16 at 19:28
  • $\begingroup$ @RemyRouyer: Just a kind of brute force. $\endgroup$ – Max Alekseyev Oct 23 '16 at 18:33

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