#Japt -h, 9 7 bytes

k ü mÊn 

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k ü mÊn :Implicit input of integer k :Prime factors ü :Group by value m :Map Ê : Length n :Sort :Implicit output of last element 

Japt -h, 9 7 bytes

k ü mÊn 

Try it

k ü mÊn :Implicit input of integer k :Prime factors ü :Group by value m :Map Ê : Length n :Sort :Implicit output of last element 

#Japt, 9 8 bytes

k ü mÊrw 

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k ü mÊrw :Implicit input of integer k  :Prime factors ü  :Group by value m  :Map Ê  : Length   r :Reduce by w : Maximum 

#Japt -h, 9 7 bytes

k ü mÊn 

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k ü mÊn :Implicit input of integer k :Prime factors ü :Group by value m :Map Ê : Length n :Sort :Implicit output of last element 

#Japt, 9 bytes

k ò¦ mÊrw 

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##Explanation

• Get prime factors with k.
• Partition on inequality with ò¦.
• Map lengths with mÊ.
• Get maximum with rw.

#Japt, 9 8 bytes

k ü mÊrw 

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k ü mÊrw :Implicit input of integer k :Prime factors ü :Group by value m :Map Ê : Length r :Reduce by w : Maximum