1,1

A number with 10*n digits could contain all ten digits(0-9) n times. The probability of this is (10n)!/((n!)^10 * 10^((10*n)-10^(10*n-1)). There are 10^10-10^(10-1/n)) numbers which are nth powers of some 10-digit numbers. So there are about (10n)!*(10^10-10^(10-1/n)))/((n!)^10 * 10^((10*n)-10^(10*n-1)) numbers which satisfy the requirements.

Fortunately, I found a larger number than those shown here, for n=26, a(n)=9160395852. Since (10n)!*(10^10-10^(10-1/n))/((n!)^10 * 10^((10*n)-10^(10*n-1)) = 0.31691419..., this is a lucky event!

Zhining Yang, Smallest Ten Digit Powers

Zhining Yang, Largest Ten Digit Powers

For n=18,a(n)=8807854905. That means 8807854905^18=101780066705166236878346559130256930813111042299349121630146805870467163387920998553309748134869974589535455566056948852225484079291126292377577073478873342409711802829742431640625 18th power of 8807854905 has all digit(0-9) each for 18 times and 8807854905 is the smallest 10-digit number which is satisified with all the rules

(Visual Basic) Function befit(ByVal s As String, ByVal num As Long) As Boolean 'tell if a string s contain all digit(0-9) for just num times Dim b(9) As Long, t As Long befit = True 'init If Len(s) <> 10 * num Then befit = False: Exit Function For i = 1 To Len(s) t = Val(Mid(s, i, 1)) b(t) = b(t) + 1 If b(t) > num Then befit = False: Exit Function Next End Function

Function mypower(ByVal num As Currency, ByVal power As Long) As String 'UDF to calculate powers of a 10-digit number Dim b(), temp ReDim b(1 To 2 * power) ReDim s(1 To 2 * power) 'The last two element of the result, i.e. num it self b(2 * power - 1) = Val(Left(num, 5)) 'init b(2 * power) = Val(Right(num, 5)) 'init For i = 2 To power temp = 0 For j = 2 * power To 1 Step -1 temp = b(j) * num + temp b(j) = Format(Val(Right(temp, 5)), "00000") '100000 adic temp = Int(temp / 10 ^ 5) Next Next mypower = Join(b, "") 'The final result End Function

Private Sub Command1_Click() Dim index As Long, j As Currency, s As String Index = CLng(InputBox("Please enter an integer within 1-30", "Info", 2)) For j = 3*Int(10 ^ (10 - 1 / index)/3) To 9999999999# Step 3 'n times 0-9 must be divisible by 3 DoEvents s = mypower(j, index) 'the result If befit(s, index) Then 's contains 0-9 each for index times Open "c:\"& index &".txt" For Binary As #1 'Output to a text file Put #1, j & "^" & index & "=" & s 'Print the result Close #1 End If End Next End Sub

Cf. A010784 A078255 A154532.

nonn,base,new

Zhining Yang (northwolves(AT)163.com), Jan 12 2009, Jan 13 2009

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 13 2009

Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Nov 01 2009