Descent to Zero

The number associated with my diurnal age today, $\mathbf{\text{27779}}$, has the interesting property that it is the smallest number that takes 22 steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". The possible concatenatable pairs and their products for 27779 are:

• 2 * 7779 = 15558
• 27 * 779 = 21033
• 277 * 79 = 21883
• 2777 * 9 = 24993

We see that 24993 is the maximum product and we repeat the process using this number as our starting point:

• 2 * 4993 = 9986
• 24 * 993 = 23832
• 249 * 93 = 23157
• 2499 * 3 = 7497

The maximum product is 23832 and so this becomes the new number and the process continues until we reach 0 after 22 steps. The progression is as follows (permalink):

27779, 24993, 23832, 19136, 11478, 9176, 6916, 5496, 5184, 4284, 3528, 2816, 1686, 1376, 988, 792, 644, 264, 128, 96, 54, 20, 0

27779 is a member of OEIS A035932: smallest number that takes $n$ steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". The initial members of this sequence are:

0, 1, 11, 26, 39, 77, 117, 139, 449, 529, 777, 1117, 2229, 2982, 4267, 4779, 5319, 5919, 8693, 12699, 14119, 17907, 27779, 47877, 80299, 103199, 135199, 274834, 293938, 312794, 606963, 653993, 773989, 1160892, 1296741, 1616696, 1986576

This permalink will check for the smallest number once the number of steps is specified. When writing the code the convention is that you add a condition to handle single-digit numbers. A common rule for sequences like this that aim to reach 0 is that single-digit numbers (other than 0) map to 0 in the next step. So looking at the above sequence we see following progressions to 0 beginning with 0 that requires zero steps (permalink):

• 0
• 1, 0
• 11, 1, 0
• 26, 12, 2, 0
• 39, 27, 14, 4, 0
• 77, 49, 36, 18, 8, 0
• 117, 77, 49, 36, 18, 8, 0
• 139, 117, 77, 49, 36, 18, 8, 0
• 449, 396, 288, 224, 88, 64, 24, 8, 0
• 529, 468, 368, 288, 224, 88, 64, 24, 8, 0
• 777, 539, 477, 329, 288, 224, 88, 64, 24, 8, 0
• 1117, 777, 539, 477, 329, 288, 224, 88, 64, 24, 8, 0
• 2229, 1998, 1862, 1116, 666, 396, 288, 224, 88, 64, 24, 8, 0
• 2982, 2378, 1896, 1728, 1376, 988, 792, 644, 264, 128, 96, 54, 20, 0