### Addendum to : sigma ( phi ( ) ) : From "5" to "5 figures"

Update to the count of solutions to the equation

```           phi ( sigma ( n ) ) = sigma ( phi ( n ) )
```

In sigma ( phi ( ) ) : From "5" to "5 figures" a table of smallest suitable values of generalized repunit primes based on Fermat prime bases is shown :

```          smallest suitable values
------ -----------------------------------------------------------------
p3     3,     7,     13,     71, 103,         541,      1091, 1367, 1627
p5     3,     7, 11, 13, 47,      127, 149, 181, 619, 929
p17    3, 5,  7, 11,     47, 71,             419
p257                   23, 59
p65537        7, 11
```

Using the values in that table , the number of known small odd solutions to

```           phi ( sigma ( n ) ) = sigma ( phi ( n ) )
```
is counted :
```cardinalities (so far)
exponents                       solutions
#p3      >=  9                  9                   >=    9
#p5      >= 10                  9 * 10              >=   90
#p17     >=  7                  9 * 10 * 7          >=  630
#p257    >=  2                  9 * 10 * 7 * 2      >= 1260
#p65537  >=  2                  9 * 10 * 7 * 2 * 2  >= 2520
----
total number of small solutions verified (so far)   >= 4509
```

More solutions have now become available :

```          smallest suitable values   updated
p3  3 7 13 71 103 541 1091 1367 1627 4177  :  9011 9551 36913 43063 49681 57917
p5  3 7 11 13 47 127 149 181 619 929 3407  :  10949 13241 13873 16519
p17  3 5 7 11 47 71 419                     :  4799
p257  23 59 487 967                          :  5657
p65537  7 11
```
In this table , the values to the right of the column of colons are not actually proven primes , but are only probable primes .
Using only proven primes , this counts the known solutions :
```cardinalities ( so far )
exponents                       solutions
#p3       = 10                  10                    =    10
#p5       = 11                  10 * 11               =   110
#p17      =  7                  10 * 11 * 7           =   770
#p257     =  4                  10 * 11 * 7 * 4       =  3080
#p65537   =  2                  10 * 11 * 7 * 4 * 2   =  6160
----
total number of small solutions verified ( so far )   = 10130
```

Counting the probable primes also , this would be the hopeful total count of probable solutions :

```cardinalities ( if proven )
exponents                       solutions
#p3       = 16                  16                    =    16
#p5       = 15                  16 * 15               =   240
#p17      =  8                  16 * 15 * 8           =  1920
#p257     =  5                  16 * 15 * 8 * 5       =  9600
#p65537   =  2                  16 * 15 * 8 * 5 * 2   = 19200
-----
total number of small solutions     ( if proven )     = 30976
```

```         limit of search for probable primes ( so far )
p3   100000
p5   100000
p17    30000
p257    10007
p65537     4423
```

Additional even solutions can be found in Elaboration of : sigma ( phi ( ) ) : From "5" to "5 figures" .

Appreciation :
Kind thanks are extended to Andy Steward ,
http://www.primes.viner-steward.org/andy/titans.html
http://www.primes.viner-steward.org/andy/annual.html
for the 1st and 2nd lines of this table , to Henri Lifchitz ,
```originally posted 2008-06-08