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2 It's Numbers 16, not Leviticus 16, that opens with the phrase in question.
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As you mentioned yourself, there are "specific reason[s] why only certain people's names go beyond the father". However, I don't think there is a hard and fast rule for each example, rather each one is brought for a specific reason.

With regards to one of your examples:

Korach's genealogy was listed in the beginning of LeviticusNumbers 16:1 only up to Levi, since Yaakov (one generation farther) specifically requested that his name not be associated with the quarrel he caused. (Rashi)

I also seem to remember a reason for listing Betzalel's genealogy (possibly due to Chur being killed for trying to prevent the Golden Calf), but I don't remember the source offhand.

As you mentioned yourself, there are "specific reason[s] why only certain people's names go beyond the father". However, I don't think there is a hard and fast rule for each example, rather each one is brought for a specific reason.

With regards to one of your examples:

Korach's genealogy was listed in the beginning of Leviticus 16:1 only up to Levi, since Yaakov (one generation farther) specifically requested that his name not be associated with the quarrel he caused. (Rashi)

I also seem to remember a reason for listing Betzalel's genealogy (possibly due to Chur being killed for trying to prevent the Golden Calf), but I don't remember the source offhand.

As you mentioned yourself, there are "specific reason[s] why only certain people's names go beyond the father". However, I don't think there is a hard and fast rule for each example, rather each one is brought for a specific reason.

With regards to one of your examples:

Korach's genealogy was listed in the beginning of Numbers 16:1 only up to Levi, since Yaakov (one generation farther) specifically requested that his name not be associated with the quarrel he caused. (Rashi)

I also seem to remember a reason for listing Betzalel's genealogy (possibly due to Chur being killed for trying to prevent the Golden Calf), but I don't remember the source offhand.

1
source | link

As you mentioned yourself, there are "specific reason[s] why only certain people's names go beyond the father". However, I don't think there is a hard and fast rule for each example, rather each one is brought for a specific reason.

With regards to one of your examples:

Korach's genealogy was listed in the beginning of Leviticus 16:1 only up to Levi, since Yaakov (one generation farther) specifically requested that his name not be associated with the quarrel he caused. (Rashi)

I also seem to remember a reason for listing Betzalel's genealogy (possibly due to Chur being killed for trying to prevent the Golden Calf), but I don't remember the source offhand.