3 added 3 characters in body

# Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&


# Explanation

FactorInteger@#


Factor the input. (e.g. 8668 -> {{2, 2}, {11, 1}, {197, 1}}; for each list in the output, the first element is the prime factor, and the second one is the power.

Join@@Table@@ ...


For each factor-power pair, duplicate the first element by the second element, and flatten the entire thing. ({{2, 2}, {11, 1}, {197, 1}} -> {{2, 2}, {11}, {197}} -> {2, 2, 11, 197})

FoldList[1##&, ... ]


Iterate through the list, multiplying the elements. ({2, 2, 11, 197} -> {2, 2 * 2, 2 * 2 * 11, 2 * 2 * 11 * 197} -> {2, 4, 44, 8668})

And@@PalindromeQ/@ ...


Check whether all of the resulting numbers are palindromes, and apply the And operator. ({2, 4, 44, 8668} -> {True, True, True, True} -> True)

# Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&


# Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&


# Explanation

FactorInteger@#


Factor the input. (e.g. 8668 -> {{2, 2}, {11, 1}, {197, 1}}; for each list in the output, the first element is the prime factor, and the second one is the power.

Join@@Table@@ ...


For each factor-power pair, duplicate the first element by the second element, and flatten the entire thing. ({{2, 2}, {11, 1}, {197, 1}} -> {{2, 2}, {11}, {197}} -> {2, 2, 11, 197})

FoldList[1##&, ... ]


Iterate through the list, multiplying the elements. ({2, 2, 11, 197} -> {2, 2 * 2, 2 * 2 * 11, 2 * 2 * 11 * 197} -> {2, 4, 44, 8668})

And@@PalindromeQ/@ ...


Check whether all of the resulting numbers are palindromes, and apply the And operator. ({2, 4, 44, 8668} -> {True, True, True, True} -> True)

2 added 3 characters in body

# Mathematica, 7171 63 bytes

Switch[#,1,True,_?PalindromeQ,#0[#And@@PalindromeQ/Max[#&@@@FactorInteger@#]],_@FoldList[1##&,False]&Join@@Table@@@FactorInteger@#]&


# Mathematica, 71 bytes

Switch[#,1,True,_?PalindromeQ,#0[#/Max[#&@@@FactorInteger@#]],_,False]&


# Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&

1

# Mathematica, 71 bytes

Switch[#,1,True,_?PalindromeQ,#0[#/Max[#&@@@FactorInteger@#]],_,False]&