Wednesday, October 07, 2020

Proportionality

In the list of consecutive positive base-ten integers: 1, 2, 3, ...

   the 1111111111st occurrence of 1 is the first 1 in 1111111111,
   the 2222222222nd occurrence of 2 is the second 2 in 2222222222,
   the 3333333333rd occurrence of 3 is the third 3 in 3333333333,
   the 4444444444th occurrence of 4 is the fourth 4 in 4444444444,
   the 5555555555th occurrence of 5 is the fifth 5 in 5555555555,
   the 6666666666th occurrence of 6 is the sixth 6 in 6666666666,
   the 7777777777th occurrence of 7 is the seventh 7 in 7777777777,
   the 8888888888th occurrence of 8 is the eighth 8 in 8888888888, and
   the 9999999999th occurrence of 9 is the ninth 9 in 9999999999.

In other bases, the n-th occurrence of 1 as the first 1 in n is given in A023037 as a base-ten equivalent. Be careful about the offset (base two is the third term). Multiply the number by d for the n-th occurrence of d as the d-th d in n, up to d = base - 1. These n-th occurrence in n assertions are special solutions to the more general problem but they are not the only ones.

4 comments:

  1. Intriguing! And what about the 11th, 111th... occurrence of 1, the 22th, 222th... occurrence of 2? And in other bases, e.g., 11[2]-th, 111[2]-th... occurrence of 1 in numbers written in base 2, etc?

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    Replies
    1. Yes, lots of room for discovery. I had actually done your 2nd, 22nd, 222nd, ... occurrences of 2: They happen in 12, 112, 612, 4112, 30612, 280182 (#2), 2615519, 24480182 (#2), 230744019, and 2222222222 (#2). I am too busy at the moment with other projects to explore any of this further.

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    2. Extending the original sequence to bases of 10, 11 etc., surely you get similar results about the AAAAA...th occurrence of A, where A means ten in base 11

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    3. For base b, there would be b-1 true assertions. In base two, the 3rd occurrence of 1 is the first 1 in 3. In base eleven, the final assertion would be that the 285311670610th occurrence of digit 10 is the tenth digit 10 in 285311670610. [Letting digit 10 = A: The AAAAAAAAAAAth occurrence of A is the Ath A in AAAAAAAAAAA.]

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