Iteration: Reverse and Subtract Maximum Digit

Today I turned 26963 days old and one of the interesting properties of this number is that it's a member of OEIS  A097155:

A097155

Numbers that reach the fixed point 89 under iteration of f(x) = reverse(x) - maxdigit(x).

In the range up to 40,000, there are only 45 numbers with this property. They are:

89, 890, 998, 2125, 3126, 5207, 6207, 7018, 7019, 8099, 8900, 9098, 9899, 9980, 10151, 10152, 10224, 12205, 12259, 12268, 14085, 14086, 15095, 15096, 17972, 18971, 21250, 22015, 22269, 23077, 24005, 24086, 24087, 25096, 26963, 27962, 30225, 31116, 31260, 33006, 33077, 33078, 34087, 35954, 36953

The remaining 39955 numbers end in zero. Here is a permalink to the calculation that generated these numbers. Let's follow the trajectory of 26963 (permalink):

26963, 36953, 35954, 45944, 44945, 54935, 53936, 63926, 62927, 72917, 71918, 81908, 80909, 90899, 99800, 890, 89

ONLY 0 AND 89 REMAIN INVARIANT
UNDER THE OPERATION OF
REVERSE AND SUBTRACT LARGEST DIGIT

Figure 1 shows the trajectory of 26963. Note how 99800 collapses to 899 when it's reversed, leading quickly to 89.

Figure 1: trajectory of 26963

Compare this trajectory to that 26962 (see What's Special About 26962?) and most other numbers (permalink):

26962, 26953, 35953, 35944, 44944, 44935, 53935, 53926, 62926, 62917, 71917, 71908, 80908, 80899, 99799, 99790, 9790, 970, 70, 0

Figure 2 shows the trajectory of 26962 which is similar to that of 26963, except that it collapses to 0.

Figure 2: trajectory of 26962

If we look at the numbers corresponding to the dates in 2023 (see Turning Dates Into Numbers), there are only two numbers that lead to in 89 and these are (permalink):

• 20230213 corresponding to the 13th February 2023
• 20231112 corresponding to the 12th November 2023

The numbers corresponding to the other 363 dates all end in zero.

In the range up to 100,000, there are 124 numbers that end in 89 making for a percentage of 0.124% (permalink). Figure 3 shows the distribution of these numbers over the range.

Figure 3