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Semi-1 number (Posted on 2024-03-13) Difficulty: 3 of 5
A positive integer n is a semi-1 number if exactly half of the integers from 1 through n contain the digit 1. For example, 16 is semi-1, because exactly 8 of the integers between 1 and 16 contain the digit 1:

{1, 10, 11, 12, 13, 14, 15, 16}.

Find the largest semi-1 number you can find!

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Computer solution | Comment 1 of 4
The largest semi-1 number I can find is 1062880.

There is an upper limit on semi-1 numbers.  The probability for an n-digit number to contain no 1s, is (0.9)^n.
ln(.5)/ln(.9) is about 6.6
(0.9)^6 ~ 0.531441
(0.9)^7 ~ 0.4782969
As numbers grow larger, the percent without a 1 goes below 50%.  So the percent which include a 1 will be higher than 50%.  From this argument it should be rare to find a semi-1 number more than 6 or 7 digits long.

2
16
24
160
270
272
1456
3398
3418
3420
3422
13120
44686
118096
674934
1062880

This is in oeis: A343839.  According to oeis, this is a finite set and these 16 numbers are the entire set.

--------------
big = 10000000
yes = 0
no = 0
for n in range(1, big+1):
    if '1' in str(n):
        yes += 1
    else:
        no += 1
    if 2*yes == n:
        print(n)

  Posted by Larry on 2024-03-13 08:12:22
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