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According to my search, the number $561=3\times11\times17$ is the only carmichael-number with $3$ prime factors, which is divisible by $11$.

Is this true ?

If yes, $11$ cannot be the smallest prime factor of a carmichael-number with $3$ prime factors. But for the other odd primes $p\le 193$ , I found a carmichael number with $3$ prime factors, $p$ being the smallest.

Can every odd prime $p\ne 11$ be the smallest prime factor of a carmichael number with $3$ prime factors ?

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Graham Jameson's "Carmichael numbers with three prime factors" might help. His web site has a link to a PDF of this document. He answers your first question "yes": there are indeed no Carmichael numbers of the form $pqr$ with $p<q<r$ and $p=11$. He gives examples for every other odd prime $p$, but only up to $p=73$.

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