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Ludic numbers

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The ludic numbers are obtained to a sieving process similar to the sieve of Eratosthenes, but with the imporant difference that crossed off numbers lose their positional attributes and change the positional attributes of the larger remaining numbers.

In the sieve of Eratosthenes implemented in its simplest form, without any optimization, some numbers may be crossed off more than once. For example, 6, 12, 18, 24, 36, 48, 54, ... (A033845) can be crossed off twice each (first for 2, then for 3); whereas with modest optimization, the algorithm skips 6 when it gets to 3, proceeding instead to 9 (the square of 3). But regardless of how many times a number is crossed off, the position of later numbers does not change.

With the ludic numbers, crossed off numbers are completely removed prior to the next step.

A. Write a range of integers in order into an array, starting with 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

B. Start with \scriptstyle p \,=\, 2 \,.

C. Circle or highlight \scriptstyle p \, and cross out its square \scriptstyle p^2 \, and all its higher multiples that haven't already been crossed off.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

C. But now remove the numbers that have been crossed off.


ILLUSTRATION OF THAT GOES HERE, FOLLOWED BY THE REST OF THE STEPS

The end result are the ludic numbers (Cf. A003309)

{1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107, 115, 119, 121, 127, 131, 143, 149, 157, 161, 173, 175, 179, 181, 193, 209, 211, ...}

Sequences

A192503 Ludic prime numbers.

{2, 3, 5, 7, 11, 13, 17, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 83, 89, 97, 107, 127, 131, 149, 157, 173, 179, 181, 193, 211, 223, 227, 233, 239, 257, 277, 283, 307, 313, ...}

A192504 Ludic non prime numbers.

{1, 25, 77, 91, 115, 119, 121, 143, 161, 175, 209, 221, 235, 247, 265, 287, 301, 329, 341, 361, 377, 407, 415, 437, 445, 475, 481, 493, 497, 517, 527, 535, 553, 565, 581, ...}

A192512 Number of ludic numbers (A003309) not greater than n.

{1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, ...}

See also

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