Super-d numbers

Single-threaded

2 - 6 takes a few seconds, 7 & 8 take a few minutes; I got tired of waiting for 9.

sub super ($d) {
    my $run = $d x $d;
    "First 10 super-{$d} numbers:\n{ (^∞ .grep: ($d * * ** $d).Str.contains($run) )[^10]}\n"
}

put .&super for 2 .. 8

Output:

First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300

Concurrent

Since waiting can be tiresome, make things faster with the race concurrency function. The only problem is that we're not sure what the stopping point is ahead of time, so when it seems safe to quit (calculate a few more result than strictly needed, since the parallel workers don't coordinate), break out of the concurrent block by raising an exception. The usual concurrency caveats apply: @super must be sized ahead of time, and only modified with an atomic operation (⚛++).

Concurrency makes little difference for d < 6, but the benefits accrue rapidly after that (greater than 10-fold speed-up for d = 8, with an 8-core CPU). However, in the end, you'll still have to wait a quite bit or the super-9 values.

sub super-d ($d,$max) {
    my $max-plus = $max + floor 2*$max/$d;
    my @super[2*$max-plus];
    {
        my $digits = $d x $d;
        my $chunk  = 250 * $d;
        my atomicint $found = 0;
        (0, 1*$chunk, 2*$chunk ... *).race(:1batch).map: -> $i {
             @super[$found⚛++] = $_ if ($_ ** $d  *  $d).contains($digits) for 1+$i .. 1+$i+$chunk;
             X::AdHoc.new.throw if $found ≥ $max-plus;
        }
        CATCH { when X::AdHoc { return @super.grep(so *).sort.head($max) } }
    }
}

say "$_: " ~ join ' ', super-d($_,10) for 2..9;

Output:

2: 19 31 69 81 105 106 107 119 127 131
3: 261 462 471 481 558 753 1036 1046 1471 1645
4: 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680
5: 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689
6: 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146
7: 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763
8: 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300
9: 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181