Self-powers numbers (SPN)

[Eric Angelini]

Hello Math-fun and Seqfan,

N=325648 is interesting if read so:

"3" means "a cube is visible in N"
    (yes, it is "8" -- 8=2*2*2)
"2" means "a square is visible in N"
    (yes, it is "25" -- 25=5*5)
    (4 is ok too, being 2*2)
"5" means "a power 5 is visible in N"
    (yes, it is "32" = 2*2*2*2*2)
"6" means "a power 6 is visible in N"
    (yes, it is "64" = 2*2*2*2*2*2)
"4" means "a power 4 is visible in N"
    (yes, it is "256" = 4*4*4*4*)
"8" means "a power 8 is visible in N"
    (yes, it is "256" =2*2*2*2*2*2*2*2)

N=832564 is a SPN too, of course.

["visible" means "as a whole": "25" is
NOT visible in 235]

Question:
Can someone compute all such SPN _which
don't include any 0's or 1's_ ?

This restriction applies because 0^a=0
and 1^b=1, which brings a lot of unwan-
ted SPN like 117 or 308:

"1" means "a power 1 is visible in N"
    (yes, it is "1" -- 1^1=1)
"7" means "a power 7 is visible in N"
    (yes, it is "1" again -- 1^7=1)

or

"3" means "a cube is visible in N"
    (yes, it is "0" -- 0*0*0=0)
"0" means "a power 0 is visible in N"
    (yes, it is "0" again -- any a^0=0)
"8" means "a power 8 is visible in N"
    (yes, it is "0" again and again
                              -- 0^8=0)
Best,
É.