Penholodigital squares

Penholodigital squares are perfect square numbers that contain all of the digits from the base in which the number is represented, except for zero, exactly once.

From the Latin prefix pene- (before, or next to, nearly)

and holo- (whole, or all)

penholodigital: Nearly-all-digits.

So, in a particular base, a penholodigital square number will contain all of the digits used in that base (except zero) once, and only once. Base eight penholodigitals contain the digit 1 through 7, base 10, 1 through 9, etc.

For example

In base 10, 139854276 is a penholodigital square. It is the square of the integer 11826, and contains every digit from 1 through 9 exactly once.

Penholodigital squares can occur in many, though not every, base. They tend to be pretty rare in lower bases.

There is a total of 1 penholodigital squares in base 2:
1² = 1

There is a total of 0 penholodigital squares in base 3:

There is a total of 0 penholodigital squares in base 4:

There is a total of 0 penholodigital squares in base 5:

There is a total of 2 penholodigital squares in base 6:
122² = 15324, 221² = 53241

There is a total of 1 penholodigital squares in base 7:
645² = 623514

There is a total of 1 penholodigital squares in base 8:
2453² = 6532471

Task

Find and display the total count, and the penholodigital squares and the integers that are squared to produce them, represented in the base in which they are calculated, for bases 9, 10, 11 and 12.

Stretch

Find and display the total count, and the penholodigital squares and the integers that are squared to produce them, represented in the base in which they are calculated, for bases 13, 14, 15, ... ?

See also

OEIS:A036744 - Penholodigital squares in base 10

Raku

(9 .. 12).map: -> $base {
    my $test = (1 ..^ $base)».base($base).join;
    my $start = $test     .parse-base($base).sqrt.Int;
    my $end   = $test.flip.parse-base($base).sqrt.Int;
    say "\nThere is a total of {+$_} penholodigital squares in base $base:\n" ~
        .map({"{.base($base)}² = {.².base($base)}"}).batch(3)».join(", ").join: "\n" given
        ($start .. $end).grep: *².base($base).comb.sort.join eq $test
}

(13 .. 16).hyper(:1batch).map: -> $base {
    my $test = (1 ..^ $base)».base($base).join;
    my $start = $test     .parse-base($base).sqrt.Int;
    my $end   = $test.flip.parse-base($base).sqrt.Int;
    my @penholo = ($start .. $end).grep: *².base($base).comb.sort.join eq $test;
    say "\nThere is a total of {+@penholo} penholodigital squares in base $base:";
    say @penholo[0,*-1].map({"{.base($base)}² = {.².base($base)}"}).batch(3)».join(", ").join: "\n" if +@penholo;
}

Output:

There is a total of 10 penholodigital squares in base 9:
3825² = 16328547, 3847² = 16523874, 4617² = 23875614
4761² = 25487631, 6561² = 47865231, 6574² = 48162537
6844² = 53184267, 7285² = 58624317, 7821² = 68573241
8554² = 82314657

There is a total of 30 penholodigital squares in base 10:
11826² = 139854276, 12363² = 152843769, 12543² = 157326849
14676² = 215384976, 15681² = 245893761, 15963² = 254817369
18072² = 326597184, 19023² = 361874529, 19377² = 375468129
19569² = 382945761, 19629² = 385297641, 20316² = 412739856
22887² = 523814769, 23019² = 529874361, 23178² = 537219684
23439² = 549386721, 24237² = 587432169, 24276² = 589324176
24441² = 597362481, 24807² = 615387249, 25059² = 627953481
25572² = 653927184, 25941² = 672935481, 26409² = 697435281
26733² = 714653289, 27129² = 735982641, 27273² = 743816529
29034² = 842973156, 29106² = 847159236, 30384² = 923187456

There is a total of 20 penholodigital squares in base 11:
42045² = 165742A893, 43152² = 173A652894, 44926² = 18792A6453
47149² = 1A67395824, 47257² = 1A76392485, 52071² = 249A758631
54457² = 2719634A85, 55979² = 286A795314, 59597² = 314672A895
632A4² = 3671A89245, 64069² = 376198A254, 68335² = 41697528A3
71485² = 46928A7153, 81196² = 5A79286413, 83608² = 632A741859
86074² = 6713498A25, 89468² = 7148563A29, 91429² = 76315982A4
93319² = 795186A234, A3A39² = 983251A764

There is a total of 23 penholodigital squares in base 12:
117789² = 135B7482A69, 16357B² = 23A5B976481, 16762B² = 24AB5379861
16906B² = 25386749BA1, 173434² = 26B859A3714, 178278² = 2835BA17694
1A1993² = 34A8125B769, 1A3595² = 354A279B681, 1B0451² = 3824B7569A1
1B7545² = 3A5B2487961, 2084A9² = 42A1583B769, 235273² = 5287BA13469
2528B5² = 5B23A879641, 25B564² = 62937B5A814, 262174² = 63A8527B194
285A44² = 73B615A8294, 29A977² = 7B9284A5361, 2A7617² = 83AB5479261
2B0144² = 8617B35A294, 307381² = 93825A67B41, 310828² = 96528AB7314
319488² = 9AB65823714, 319A37² = 9B2573468A1

There is a total of 0 penholodigital squares in base 13:

There is a total of 160 penholodigital squares in base 14:
1129535² = 126A84D79C53B, 3A03226² = DB3962A7541C8

There is a total of 419 penholodigital squares in base 15:
4240C58² = 12378DA5B6EC94, EE25E4A² = ED4C93285671BA

There is a total of 740 penholodigital squares in base 16:
11156EB6² = 123DA7F85BCE964, 3FD8F786² = FEC81B69573DA24