All rational numbers can be put into a 1-to-1 matching with the integers. For the positive of each, the below table shows a 1-to-1 matching where the rationals are laid out in order of the total of the numerator and denominator of the number expressed as a fraction in simplest (reduced) form. Within one total of numerator and denominator, they are in order of the numerator.

    1     1              1/14    64
  1/2     2              2/13    65
    2     3              4/11    66
  1/3     4               7/8    67
    3     5               8/7    68
  1/4     6              11/4    69
  2/3     7              13/2    70
  3/2     8                14    71
    4     9              1/15    72
  1/5    10              3/13    73
    5    11              5/11    74
  1/6    12               7/9    75
  2/5    13               9/7    76
  3/4    14              11/5    77
  4/3    15              13/3    78
  5/2    16                15    79
    6    17              1/16    80
  1/7    18              2/15    81
  3/5    19              3/14    82
  5/3    20              4/13    83
    7    21              5/12    84
  1/8    22              6/11    85
  2/7    23              7/10    86
  4/5    24               8/9    87
  5/4    25               9/8    88
  7/2    26              10/7    89
    8    27              11/6    90
  1/9    28              12/5    91
  3/7    29              13/4    92
  7/3    30              14/3    93
    9    31              15/2    94
 1/10    32                16    95
  2/9    33              1/17    96
  3/8    34              5/13    97
  4/7    35              7/11    98
  5/6    36              11/7    99
  6/5    37              13/5   100
  7/4    38                17   101
  8/3    39              1/18   102
  9/2    40              2/17   103
   10    41              3/16   104
 1/11    42              4/15   105
  5/7    43              5/14   106
  7/5    44              6/13   107
   11    45              7/12   108
 1/12    46              8/11   109
 2/11    47              9/10   110
 3/10    48              10/9   111
  4/9    49              11/8   112
  5/8    50              12/7   113
  6/7    51              13/6   114
  7/6    52              14/5   115
  8/5    53              15/4   116
  9/4    54              16/3   117
 10/3    55              17/2   118
 11/2    56                18   119
   12    57              1/19   120
 1/13    58              3/17   121
 3/11    59              7/13   122
  5/9    60              9/11   123
  9/5    61              11/9   124
 11/3    62              13/7   125
   13    63              17/3   126
                           19   127

Using this scheme, what rational number is matched to the integer 1 million?

When 1 million is considered as a rational number, to what integer is it matched?

Good thing I can get a good estimate with ZERO programming, just with a little help from the OEIS.

Posted by Brian Smith on 2024-05-20 11:28:30