A Note on Carmichael Numbers of the form $N=(d+1)(3d+1)(4d+1)$

We study integers of the form $$N=(d+1)(3d+1)(4d+1)\mathrm{.}$$ Using Korselt’s criterion, we prove that, provided the three factors are prime, $N$ is a Carmichael number if and only if $$d\equiv 4(\mathrm{m}\mathrm{o}\mathrm{d}12)\mathrm{.}$$ We also show that the special case $d=2^m$ produces only $1105$ under the prime-factor hypothesis.