Need help with sequences

[Tanya Khovanova]

Hello Seqfans,

I am not very good with Mathematica and my computer is slow. I would like to suggest several sequences for which I have only two terms. If you are interested, please, calculate more and submit.

Number 1:
4624, 595968 - Numbers n such that the sum of the 4 to the powers of its digits equals n.
Comment: If you use 2 instead of 4, such numbers do not exist. If you use 3, the only number is 12.
Example: 4^4 + 4^6 + 4^2 + 4^4 = 256 + 4096 + 16 + 256 = 4624.
Math: Select[Range[10000000], Plus @@ (4^IntegerDigits[#]) == # &]

Number 2:
5986, 8614 - Numbers that together with their prime factors contain every non-zero digit exactly once.
Example: 5986 = 2*41*73.
Math: Select[Range[2, 1000000], 
  Sort[Join[IntegerDigits[#], 
          Flatten[IntegerDigits[Transpose[FactorInteger[#]][[1]]]]]] == {1, 2, 3, 4, 5, 6, 7, 8, 9} &]

Number 3:
10, 2328 Numbers n such that prime(prime(n)) starts with n.
Example: prime(10) = 29, prime(29) = 109, which starts with 10.
Math: Select[Range[100000], 
  Prime[Prime[#]] - #*10^(Length[IntegerDigits[Prime[Prime[#]]]] - 
                Length[IntegerDigits[#]]) == 
      Mod[Prime[Prime[#]], 
        10^(Length[IntegerDigits[Prime[Prime[#]]]] - 
              Length[IntegerDigits[#]])] &]
Compare: A067928  Numbers n such that the digits of the n-th prime 
begin with n.; A074978  Prime(prime(n)) ends with n.