user
Definition

An   **eban**   number is a number that has no letter    **e**    in it when the number is spelled in English.

Or more literally,   spelled numbers that contain the letter   **e**   are banned.

The American version of spelling numbers will be used here   (as opposed to the British).

**2,000,000,000**   is two billion,   *not*   two milliard.

Only numbers less than   **one sextillion**   (**1021**)   will be considered in/for this task.

This will allow optimizations to be used.

Task
:   :   :   * show all eban numbers   ≤   **1,000**   (in a horizontal format),   and a count
            * show all eban numbers between   **1,000**   and   **4,000**   (inclusive),   and a count
            * show a count of all eban numbers up and including           **10,000**
            * show a count of all eban numbers up and including         **100,000**
            * show a count of all eban numbers up and including      **1,000,000**
            * show a count of all eban numbers up and including    **10,000,000**
            * show all output here.

See also
:   * The MathWorld entry:   eban numbers.
    * The OEIS entry:   A6933, eban numbers.
    * Number names.
assistant
(formerly Perl 6)

**Works with**: Rakudo version 2018.12

Modular approach, very little is hard coded. Change the $upto order-of-magnitude limit to adjust the search/display ranges. Change the letter(s) given to the enumerate / count subs to modify which letter(s) to disallow.

Will handle multi-character 'bans'. Demonstrate for e-ban, t-ban and subur-ban.

Directly find :

:   * OEIS:A006933 Numbers without e: Eban
    * OEIS:A008521 Numbers without o: Oban
    * OEIS:A008523 Numbers without t: Tban
    * OEIS:A072954 Numbers without a, i, l, t: TALIban
    * OEIS:A072955 Numbers without b, r, s, u: SUBURban
    * OEIS:A072956 Numbers without r, t, u: TURban
    * OEIS:A072957 Numbers without r, u: URban
    * OEIS:A072958 Numbers without a, c, i, l: CALIban
    * OEIS:A089589 Numbers without i: Iban
    * OEIS:A089590 Numbers without u: Uban
    * *and so on...*

Considering numbers up to **1021**, as the task directions suggest.

```
use Lingua::EN::Numbers;

sub nban ($seq, $n = 'e') { ($seq).map: { next if .&cardinal.contains(any($n.lc.comb)); $_ } }

sub enumerate ($n, $upto) {
    my @ban = [nban(1 .. 99, $n)],;
    my @orders;
    (2 .. $upto).map: -> $o {
        given $o % 3 { # Compensate for irregulars: 11 - 19
            when 1  { @orders.push: [flat (10**($o - 1) X* 10 .. 19).map(*.&nban($n)), |(10**$o X* 2 .. 9).map: *.&nban($n)] }
            default { @orders.push: [flat (10**$o X* 1 .. 9).map: *.&nban($n)] }
        }
    }
    ^@orders .map: -> $o {
        @ban.push: [] and next unless +@orders[$o];
        my @these;
        @orders[$o].map: -> $m {
            @these.push: $m;
            for ^@ban -> $b {
                next unless +@ban[$b];
                @these.push: $_ for (flat @ban[$b]) »+» $m ;
            }
        }
        @ban.push: @these;
    }
    @ban.unshift(0) if nban(0, $n);
    flat @ban.map: *.flat;
}

sub count ($n, $upto) {
    my @orders;
    (2 .. $upto).map: -> $o {
        given $o % 3 { # Compensate for irregulars: 11 - 19
            when 1  { @orders.push: [flat (10**($o - 1) X* 10 .. 19).map(*.&nban($n)), |(10**$o X* 2 .. 9).map: *.&nban($n)] }
            default { @orders.push: [flat (10**$o X* 1 .. 9).map: *.&nban($n)] }
        }
    }
    my @count  = +nban(1 .. 99, $n);
    ^@orders .map: -> $o {
        @count.push: 0 and next unless +@orders[$o];
        my $prev = so (@orders[$o].first( { $_ ~~ /^ '1' '0'+ $/ } ) // 0 );
        my $sum = @count.sum;
        my $these = +@orders[$o] * $sum + @orders[$o];
        $these-- if $prev;
        @count[1 + $o] += $these;
        ++@count[$o]  if $prev;
    }
    ++@count[0] if nban(0, $n);
    [\+] @count;
}

#for < e o t tali subur tur ur cali i u > -> $n { # All of them
for < e t subur > -> $n { # An assortment for demonstration
    my $upto   = 21; # 1e21
    my @bans   = enumerate($n, 4);
    my @counts = count($n, $upto);

    # DISPLAY
    my @k = @bans.grep: * < 1000;
    my @j = @bans.grep: 1000 <= * <= 4000;
    put "\n============= {$n}-ban: =============\n" ~
        "{$n}-ban numbers up to 1000: {+@k}\n[{@k».&comma}]\n\n" ~
        "{$n}-ban numbers between 1,000 & 4,000: {+@j}\n[{@j».&comma}]\n" ~
        "\nCounts of {$n}-ban numbers up to {cardinal 10**$upto}"
        ;

    my $s = max (1..$upto).map: { (10**$_).&cardinal.chars };
    @counts.unshift: @bans.first: * > 10, :k;
    for ^$upto -> $c {
        printf "Up to and including %{$s}s: %s\n", cardinal(10**($c+1)), comma(@counts[$c]);
    }
}

```

Output:

```
============= e-ban: =============
e-ban numbers up to 1000: 19
[2 4 6 30 32 34 36 40 42 44 46 50 52 54 56 60 62 64 66]

e-ban numbers between 1,000 & 4,000: 21
[2,000 2,002 2,004 2,006 2,030 2,032 2,034 2,036 2,040 2,042 2,044 2,046 2,050 2,052 2,054 2,056 2,060 2,062 2,064 2,066 4,000]

Counts of e-ban numbers up to one sextillion
Up to and including                     ten: 3
Up to and including             one hundred: 19
Up to and including            one thousand: 19
Up to and including            ten thousand: 79
Up to and including    one hundred thousand: 399
Up to and including             one million: 399
Up to and including             ten million: 1,599
Up to and including     one hundred million: 7,999
Up to and including             one billion: 7,999
Up to and including             ten billion: 31,999
Up to and including     one hundred billion: 159,999
Up to and including            one trillion: 159,999
Up to and including            ten trillion: 639,999
Up to and including    one hundred trillion: 3,199,999
Up to and including         one quadrillion: 3,199,999
Up to and including         ten quadrillion: 12,799,999
Up to and including one hundred quadrillion: 63,999,999
Up to and including         one quintillion: 63,999,999
Up to and including         ten quintillion: 255,999,999
Up to and including one hundred quintillion: 1,279,999,999
Up to and including          one sextillion: 1,279,999,999

============= t-ban: =============
t-ban numbers up to 1000: 56
[0 1 4 5 6 7 9 11 100 101 104 105 106 107 109 111 400 401 404 405 406 407 409 411 500 501 504 505 506 507 509 511 600 601 604 605 606 607 609 611 700 701 704 705 706 707 709 711 900 901 904 905 906 907 909 911]

t-ban numbers between 1,000 & 4,000: 0
[]

Counts of t-ban numbers up to one sextillion
Up to and including                     ten: 7
Up to and including             one hundred: 9
Up to and including            one thousand: 56
Up to and including            ten thousand: 56
Up to and including    one hundred thousand: 56
Up to and including             one million: 57
Up to and including             ten million: 392
Up to and including     one hundred million: 785
Up to and including             one billion: 5,489
Up to and including             ten billion: 38,416
Up to and including     one hundred billion: 76,833
Up to and including            one trillion: 537,824
Up to and including            ten trillion: 537,824
Up to and including    one hundred trillion: 537,824
Up to and including         one quadrillion: 537,825
Up to and including         ten quadrillion: 3,764,768
Up to and including one hundred quadrillion: 7,529,537
Up to and including         one quintillion: 52,706,752
Up to and including         ten quintillion: 52,706,752
Up to and including one hundred quintillion: 52,706,752
Up to and including          one sextillion: 52,706,752

============= subur-ban: =============
subur-ban numbers up to 1000: 35
[1 2 5 8 9 10 11 12 15 18 19 20 21 22 25 28 29 50 51 52 55 58 59 80 81 82 85 88 89 90 91 92 95 98 99]

subur-ban numbers between 1,000 & 4,000: 0
[]

Counts of subur-ban numbers up to one sextillion
Up to and including                     ten: 6
Up to and including             one hundred: 35
Up to and including            one thousand: 35
Up to and including            ten thousand: 35
Up to and including    one hundred thousand: 35
Up to and including             one million: 36
Up to and including             ten million: 216
Up to and including     one hundred million: 2,375
Up to and including             one billion: 2,375
Up to and including             ten billion: 2,375
Up to and including     one hundred billion: 2,375
Up to and including            one trillion: 2,375
Up to and including            ten trillion: 2,375
Up to and including    one hundred trillion: 2,375
Up to and including         one quadrillion: 2,375
Up to and including         ten quadrillion: 2,375
Up to and including one hundred quadrillion: 2,375
Up to and including         one quintillion: 2,375
Up to and including         ten quintillion: 2,375
Up to and including one hundred quintillion: 2,375
Up to and including          one sextillion: 2,375
```

Note that the limit to one sextillion is somewhat arbitrary and is just to match the task parameters.

This will quite happily count \*-bans up to one hundred centillion. (10305) It takes longer, but still on the order of seconds, not minutes.

```
Counts of e-ban numbers up to one hundred centillion
 ...
Up to and including one hundred centillion: 35,184,372,088,831,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999
```