Pyth, 24 bytes

FkrPE8=Y+Y^@k1@k0;<eSYhE

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Explanation

Fk    For loop with the iterator as k
E     Take first input.
PE    List of prime factors
rPE8  Run length encoding over the list, eg: [2, 2, 3] becomes [[2, 2], [1, 3]]
Y     Initialized to an empty list by default.
@k1   Element at index 1 of k.
@k0   Element at index 0 of k.
^     Apply exponentiation.
=Y+Y  Add element to Y
SY    Sort Y
e     Last element
hE    Take next input and increment by 1
<     Is it less than?

Pyth, 24 23 bytes

FkrPE8 aY^@k1@k0;<eSYhE

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Explanation

Fk    For loop with the iterator as k
E     Take first input.
PE    List of prime factors
rPE8  Run length encoding over the list, eg: [2, 2, 3] becomes [[2, 2], [1, 3]]
Y     Initialized to an empty list by default.
@k1   Element at index 1 of k.
@k0   Element at index 0 of k.
^     Apply exponentiation.
aY    Append element to Y
SY    Sort Y
e     Last element
hE    Take next input and increment by 1
<     Is it less than?

Pyth, 24 bytes

FkrPE8=Y+Y^@k1@k0;<eSYhE

Try it online!

Pyth, 24 bytes

FkrPE8=Y+Y^@k1@k0;<eSYhE

Try it online!

Explanation

Fk    For loop with the iterator as k
E     Take first input.
PE    List of prime factors
rPE8  Run length encoding over the list, eg: [2, 2, 3] becomes [[2, 2], [1, 3]]
Y     Initialized to an empty list by default.
@k1   Element at index 1 of k.
@k0   Element at index 0 of k.
^     Apply exponentiation.
=Y+Y  Add element to Y
SY    Sort Y
e     Last element
hE    Take next input and increment by 1
<     Is it less than?