user
A **Curzon number** is defined to be a positive integer **n** for which **2n + 1** is evenly divisible by **2 × n + 1**.

**Generalized Curzon numbers** are those where the positive integer **n**, using a base integer **k**, satisfy the condition that **kn + 1** is evenly divisible by **k × n + 1**.

*Base here does not imply the radix of the counting system; rather the integer the equation is based on. All calculations should be done in base 10.*

Generalized Curzon numbers only exist for even base integers.

Task

* Find and show the first **50 Generalized Curzon numbers** for even base integers from **2** through **10**.

Stretch

* Find and show the **one thousandth**.

See also

* Numbers Aplenty - Curzon numbers * OEIS:A224486 - Numbers k such that 2\*k+1 divides 2^k+1 (Curzon numbers)

*and even though it is not specifically mentioned that they are Curzon numbers:*

* OEIS:A230076 - (A007521(n)-1)/4 (Generalized Curzon numbers with a base 4)
assistant
```
sub curzon ($base) { lazy (1..∞).hyper.map: { $_ if (exp($_, $base) + 1) %% ($base × $_ + 1) } };

for <2 4 6 8 10> {
    my $curzon = .&curzon;
    say "\nFirst 50 Curzon numbers using a base of $_:\n" ~
      $curzon[^50].batch(10)».fmt("%4s").join("\n") ~
      "\nOne thousandth: " ~ $curzon[999]
}

```

Output:

```
First 50 Curzon numbers using a base of 2:
   1    2    5    6    9   14   18   21   26   29
  30   33   41   50   53   54   65   69   74   78
  81   86   89   90   98  105  113  114  125  134
 138  141  146  153  158  165  173  174  186  189
 194  198  209  210  221  230  233  245  249  254
One thousandth: 8646

First 50 Curzon numbers using a base of 4:
   1    3    7    9   13   15   25   27   37   39
  43   45   49   57   67   69   73   79   87   93
  97   99  105  115  127  135  139  153  163  165
 169  175  177  183  189  193  199  205  207  213
 219  235  249  253  255  265  267  273  277  279
One thousandth: 9375

First 50 Curzon numbers using a base of 6:
   1    6   30   58   70   73   90  101  105  121
 125  146  153  166  170  181  182  185  210  233
 241  242  266  282  290  322  373  381  385  390
 397  441  445  446  450  453  530  557  562  585
 593  601  602  605  606  621  646  653  670  685
One thousandth: 20717

First 50 Curzon numbers using a base of 8:
   1   14   35   44   72   74   77  129  131  137
 144  149  150  185  200  219  236  266  284  285
 299  309  336  357  381  386  390  392  402  414
 420  441  455  459  470  479  500  519  527  536
 557  582  600  602  617  639  654  674  696  735
One thousandth: 22176

First 50 Curzon numbers using a base of 10:
   1    9   10   25  106  145  190  193  238  253
 306  318  349  385  402  462  486  526  610  649
 658  678  733  762  810  990  994 1033 1077 1125
1126 1141 1149 1230 1405 1422 1441 1485 1509 1510
1513 1606 1614 1630 1665 1681 1690 1702 1785 1837
One thousandth: 46845
```