This finds your sample, 82000 in a split second:

#! perl -slw
use strict;

sub gen{
    my( $n, $p ) = @_;
    return 1 if $p == 0;
    my @n = gen( $n, $p - 1 );
    my $v = $n**$p;
    return @n, $v, map{ $v + $_ } @n;
}

our $LIMIT //= 10;
our $N //= 5;

my %tests;
for( 3 .. $N ) {
    ++$tests{ $_ } for gen( $_, $LIMIT );
}

$tests{ $_ } == $N-2 and print for sort{ $a <=> $b } keys %tests;

__END__
C:\test>1130506
1
82000
[download]

To go much higher you'd probably have to convert the recursive generator to an iterator to avoid memory problems.