From: Jon Schoenfield (jonscho(AT)hiwaay.net)
Subject: Re: A121319
Date: Tue, 24 Apr 2007 00:43:24 -0500

==========

The following Excel/VBA macro code was used to compute values for OEIS 
A121319 (under the assumption that A121319(n) must have at least n actual 
digits, so, e.g., the 14-digit number 75353432948736 cannot serve as 
A121319(15), even though the last 15 digits of 2^75353432948736 are 
075353432948736).

The approach used is based on the following:

- It is easily shown that a(n) mod 100 = 36 for all n > 1.

- Thus a(3) is the smallest value from the set {136, 236, 336, ...} for 
which a(3) mod 1000 = (2^a(3)) mod 1000; a(3) is found to be 736.

- It is then easily shown that a(n) mod 1000 = 736 for all n > 2.

- Thus a(4) is the smallest value from the set {1736, 2736, 3736 ...} for 
which a(4) mod 10000 = (2^a(4)) mod 10000; a(4) is found to be 8376 ....

The code was not intended for publication, and is included only at the 
request of Neil Sloane, who has assured me that no one will reproach me for 
"inelegant programming."  (I'm afraid "inelegant" would be putting it 
mildly!)  If you find in the code below (or in the logical approach 
described above) any errors that actually lead to incorrect output (i.e., to 
the writing of incorrect values to the cells of the Excel spreadsheet), 
please let me know.  Please accept my apology for the various things about 
the code that are inelegant, inefficient, etc.

-- Jon E. Schoenfield

==========

Option Explicit

Sub Macro1()

Dim lK As Long
Dim lNMax As Long
Dim lNLastDigitsCompute As Long
Dim lNLastDigitsMatch As Long
Dim strTemp As String
Dim lN As Long
Dim lIDigit As Long
Dim lJDigit As Long
Dim lI As Long

Dim bLastNDigitsMatch As Boolean

lNMax = 80
lNLastDigitsCompute = lNMax

ReDim lDigitsTwoToPowerOf10(lNMax, lNLastDigitsCompute) As Long
ReDim lDigitsTwoToKthPower(lNMax, lNLastDigitsCompute) As Long
ReDim lDigitsK(lNMax, lNLastDigitsCompute) As Long

Columns("A:B").Clear

lDigitsTwoToPowerOf10(0, 0) = 2  ' 2^(10^0) = 2

For lN = 1 To lNMax

   ReDim lDigitsMultiplier(lNLastDigitsCompute) As Long

   For lIDigit = 0 To lNLastDigitsCompute
      lDigitsMultiplier(lIDigit) = lDigitsTwoToPowerOf10(lN - 1, lIDigit)
   Next lIDigit

   ReDim lDigitsProduct(lNLastDigitsCompute) As Long
   lDigitsProduct(0) = 1 ' init

   For lI = 1 To 10 ' multiply 10 times
      Mult lDigitsProduct(), lDigitsMultiplier(), lNLastDigitsCompute
   Next lI

   ' copy result
   For lIDigit = 0 To lNLastDigitsCompute
      lDigitsTwoToPowerOf10(lN, lIDigit) = lDigitsProduct(lIDigit)
   Next lIDigit

Next lN


lDigitsK(1, 0) = 4 ' ones digit
lDigitsK(1, 1) = 1 ' tens digit

lDigitsK(2, 0) = 6 ' ones digit
lDigitsK(2, 1) = 3 ' tens digit

For lN = 1 To 2

   ReDim lDigitsProduct(lNLastDigitsCompute) As Long
   lDigitsProduct(0) = 1 ' init

   For lJDigit = 0 To lNLastDigitsCompute

      ReDim lDigitsMultiplier(lNLastDigitsCompute) As Long
      For lIDigit = 0 To lNLastDigitsCompute
         lDigitsMultiplier(lIDigit) = lDigitsTwoToPowerOf10(lJDigit, 
lIDigit)
      Next lIDigit

      For lI = 1 To lDigitsK(lN, lJDigit)
         Mult lDigitsProduct(), lDigitsMultiplier(), lNLastDigitsCompute
      Next lI

   Next lJDigit

   ' copy result
   For lIDigit = 0 To lNLastDigitsCompute
      lDigitsTwoToKthPower(lN, lIDigit) = lDigitsProduct(lIDigit)
   Next lIDigit

   Cells(lN, 1).Value = lN

   ' get string for K
   strTemp = ""

   For lIDigit = lNLastDigitsCompute To 0 Step -1
      strTemp = strTemp & LTrim(RTrim(Str(lDigitsK(lN, lIDigit))))
      If strTemp = "0" Then
         strTemp = ""
      End If
   Next lIDigit

   Cells(lN, 2).Value = "'" & strTemp

Next lN

For lN = 3 To lNMax

   ' Init K by copying previous value
   For lIDigit = 0 To lNLastDigitsCompute
      lDigitsK(lN, lIDigit) = lDigitsK(lN - 1, lIDigit)
   Next lIDigit

   ' Init 2^K by copying previous value
   For lIDigit = lNLastDigitsCompute To 0 Step -1
      lDigitsTwoToKthPower(lN, lIDigit) = lDigitsTwoToKthPower(lN - 1, 
lIDigit)
   Next lIDigit

   ' Do the following loop until the last lN digits match

   Do
      bLastNDigitsMatch = True

      For lIDigit = lNLastDigitsCompute To 0 Step -1
         If lDigitsK(lN, lIDigit) <> 0 Then ' most significant digit of K
            If lIDigit < lN - 1 Then ' K actually has fewer than lN digits
               bLastNDigitsMatch = False
            End If
            Exit For
         End If
      Next lIDigit

      If bLastNDigitsMatch Then
         For lIDigit = lN - 1 To 0 Step -1
            If lDigitsK(lN, lIDigit) <> lDigitsTwoToKthPower(lN, lIDigit) 
Then
               bLastNDigitsMatch = False
               Exit For
            End If
         Next lIDigit
      End If

      If bLastNDigitsMatch Then
         Exit Do
      Else
         ' Increment K
         ' (E.g., at lN = 3, we're looking for a value of K whose last 3 
digits
         '  match the last 3 digits of 2^K; the last 2 digits currently 
match,
         '  so try incrementing K by 100.)
         '
         lDigitsK(lN, lN - 1) = lDigitsK(lN, lN - 1) + 1
         lIDigit = lN - 1
         Do While lDigitsK(lN, lIDigit) > 9
            lDigitsK(lN, lIDigit) = lDigitsK(lN, lIDigit) - 10
            lIDigit = lIDigit + 1
            lDigitsK(lN, lIDigit) = lDigitsK(lN, lIDigit) + 1
         Loop

         ' Update 2^K

         ReDim lDigitsProduct(lNLastDigitsCompute) As Long
         For lIDigit = 0 To lNLastDigitsCompute
            lDigitsProduct(lIDigit) = lDigitsTwoToKthPower(lN, lIDigit)
         Next lIDigit

         ReDim lDigitsMultiplier(lNLastDigitsCompute) As Long
         For lIDigit = 0 To lNLastDigitsCompute
            lDigitsMultiplier(lIDigit) = lDigitsTwoToPowerOf10(lN - 1, 
lIDigit)
         Next lIDigit

         Mult lDigitsProduct(), lDigitsMultiplier(), lNLastDigitsCompute

         ' copy result
         For lIDigit = 0 To lNLastDigitsCompute
            lDigitsTwoToKthPower(lN, lIDigit) = lDigitsProduct(lIDigit)
         Next lIDigit

      End If

   Loop

   Cells(lN, 1).Value = lN

   ' get string for K
   strTemp = ""

   For lIDigit = lNLastDigitsCompute To 0 Step -1
      strTemp = strTemp & LTrim(RTrim(Str(lDigitsK(lN, lIDigit))))
      If strTemp = "0" Then
         strTemp = ""
      End If
   Next lIDigit

   Cells(lN, 2).Value = "'" & strTemp

Next lN

End Sub

Sub Mult(lDigitsProduct() As Long, lDigitsMultiplier() As Long, 
lNLastDigitsCompute As Long)

Dim lIDigit As Long
Dim lJDigit As Long
Dim lSum As Long

ReDim lDigitsTempProduct(lNLastDigitsCompute) As Long

lSum = 0

For lIDigit = 0 To lNLastDigitsCompute
   For lJDigit = 0 To lIDigit
      lSum = lSum + lDigitsProduct(lIDigit - lJDigit) * 
lDigitsMultiplier(lJDigit)
   Next lJDigit
   lDigitsTempProduct(lIDigit) = lSum Mod 10
   lSum = lSum \ 10
Next lIDigit

' Copy resulting product into the lDigitsProduct() array
For lIDigit = 0 To lNLastDigitsCompute
   lDigitsProduct(lIDigit) = lDigitsTempProduct(lIDigit)
Next lIDigit

End Sub


--------------------------
Email from njas(AT)research.att.com:

One very good reason for making the program you
used public is that enables people to check your
work!   So I would like to get
the actual program you used, rather than a cleaned-up
version.  

Just put a sentence at the beginning saying that
the program was not intended for publication, and is included
only at my request!

Neil