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Definitions

Let **n** be a positive integer and **l(n)** be the number of its digits in base **b**.

Express **n** as the product of its prime factors raised to the appropriate powers. Let **D(n)** be the total number of its base **b** digits in all its prime factors and in all their exponents that are greater than 1.

Then **n** is defined to be:

1. a wasteful (or extravagant) number if l(n) **<** D(n); or

2. an equidigital number if l(n) **=** D(n); or

3. a frugal (or economical) number if l(n) **>** D(n)

in base **b**.

By convention, the number **1** is considered to be an **equidigital** number in **any** base even though it has no prime factors.

For the avoidance of any doubt, the number **0** is not a positive integer (and arguably not a natural number either) and so is excluded from all 3 categories.

An **economical** number is sometimes defined as being one for which l(n) **>=** D(n) though this usage won't be followed here.

Examples

In base 10, the number 30 has a prime factorization of 2 x 3 x 5. The total number of digits is 3 (all exponents being 1) which is more than the 2 digits 30 has. So 30 is wasteful in base 10.

In base 10, the number 49 has a prime factorization of 7². The total number of digits, including those of the exponent, is 2 which is the same as the 2 digits 49 has. So 49 is equidigital in base 10.

In base 10, the number 125 has a prime factorization of 5³. The total number of digits, including those of the exponent, is 2 which is less than the 3 digits 125 has. So 125 is frugal in base 10.

In base 2, the number 100000 (32 decimal) has a prime factorization of 10^101 (2^5 decimal). The total number of binary digits, including those of the exponent, is 5 which is less than the 6 binary digits 100000 has. So 32 is frugal in base 2 (but equidigital in base 10).

Task

Compute and show here the first **50** and the **10,000th** number in base **10** for each of the three categories of number defined above.

Also compute and show how many numbers less than **1,000,000** fall into each of the three categories.

Bonus

Do the same for base **11**, but show the results in base **10**.

References

* OEIS: A046760 - Wasteful numbers
* OEIS: A046758 - Equidigital numbers
* OEIS: A046759 - Economical numbers
assistant
```
use Prime::Factor;
use Lingua::EN::Numbers;

my %cache;

sub factor-char-sum ($n, $base = 10) { sum $n.&prime-factors.Bag.map: { .key.base($base).chars + (.value > 1 ?? .value.base($base).chars !! 0) } }

sub economical  ($n, $base = 10) { ($n >  1) && $n.base($base).chars >  (%cache{$base}[$n] //= factor-char-sum $n, $base) }
sub equidigital ($n, $base = 10) { ($n == 1) || $n.base($base).chars == (%cache{$base}[$n] //= factor-char-sum $n, $base) }
sub extravagant ($n, $base = 10) {              $n.base($base).chars <  (%cache{$base}[$n] //= factor-char-sum $n, $base) }


for 10, 11 -> $base {
    %cache{$base}[3e6] = Any; # preallocate to avoid concurrency issues
    say "\nIn Base $base:";
    for &extravagant, &equidigital, &economical -> &sub {
        say "\nFirst 50 {&sub.name} numbers:";
        say (^∞).grep( {.&sub($base)} )[^50].batch(10)».&comma».fmt("%6s").join: "\n";
        say "10,000th: " ~ (^∞).hyper(:2000batch).grep( {.&sub($base)} )[9999].&comma;
    }

    my $upto = 1e6.Int;
    my atomicint ($extravagant, $equidigital, $economical);
    say "\nOf the positive integers up to {$upto.&cardinal}:";
    (1..^$upto).race(:5000batch).map: { .&extravagant($base) ?? ++⚛$extravagant !! .&equidigital($base) ?? ++⚛$equidigital !! ++⚛$economical };
    say " Extravagant: {comma $extravagant}\n Equidigital: {comma $equidigital}\n  Economical: {comma $economical}";
    %cache{$base} = Empty;
}

```

Output:

```
In Base 10:

First 50 extravagant numbers:
     4      6      8      9     12     18     20     22     24     26
    28     30     33     34     36     38     39     40     42     44
    45     46     48     50     51     52     54     55     56     57
    58     60     62     63     65     66     68     69     70     72
    74     75     76     77     78     80     82     84     85     86
10,000th: 14,346

First 50 equidigital numbers:
     1      2      3      5      7     10     11     13     14     15
    16     17     19     21     23     25     27     29     31     32
    35     37     41     43     47     49     53     59     61     64
    67     71     73     79     81     83     89     97    101    103
   105    106    107    109    111    112    113    115    118    119
10,000th: 33,769

First 50 economical numbers:
   125    128    243    256    343    512    625    729  1,024  1,029
 1,215  1,250  1,280  1,331  1,369  1,458  1,536  1,681  1,701  1,715
 1,792  1,849  1,875  2,048  2,187  2,197  2,209  2,401  2,560  2,809
 3,125  3,481  3,584  3,645  3,721  4,096  4,374  4,375  4,489  4,802
 4,913  5,041  5,103  5,329  6,241  6,250  6,561  6,859  6,889  7,203
10,000th: 1,953,125

Of the positive integers up to one million:
 Extravagant: 831,231
 Equidigital: 165,645
  Economical: 3,123

In Base 11:

First 50 extravagant numbers:
     4      6      8      9     10     12     18     20     22     24
    26     28     30     33     34     36     38     39     40     42
    44     45     46     48     50     51     52     54     55     56
    57     58     60     62     63     65     66     68     69     70
    72     74     75     76     77     78     80     82     84     85
10,000th: 12,890

First 50 equidigital numbers:
     1      2      3      5      7     11     13     14     15     16
    17     19     21     23     25     27     29     31     32     35
    37     41     43     47     49     53     59     61     64     67
    71     73     79     81     83     89     97    101    103    107
   109    113    121    122    123    127    129    131    133    134
10,000th: 33,203

First 50 economical numbers:
   125    128    243    256    343    512    625    729  1,024  1,331
 1,369  1,458  1,536  1,681  1,701  1,715  1,792  1,849  1,875  2,048
 2,187  2,197  2,209  2,401  2,560  2,809  3,072  3,125  3,481  3,584
 3,645  3,721  4,096  4,374  4,375  4,489  4,802  4,913  5,041  5,103
 5,120  5,329  6,241  6,250  6,561  6,859  6,889  7,168  7,203  7,921
10,000th: 2,659,171

Of the positive integers up to one million:
 Extravagant: 795,861
 Equidigital: 200,710
  Economical: 3,428
```