Digitally Distinct (2D) and Doubly Digitally Distinct (3D) Numbers

A Digitally Distinct Number or 2D number is a term that I concocted to describe a number that has:

• no repeated digits  
• an additive digital root that is different to any of its digits

The number associated with my diurnal age today, 26870, is one such number since it clearly has no repeated digits and its additive digital root is 5.

The numbers 0 to 9 do not qualify because they are identical to their additive digital roots. However, 12 has an additive digital root of 3 and thus it is the first 2D number and begins a run of seven consecutive such numbers viz. 12, 13, 14, 15, 16, 17 and 18. The percentage of such numbers declines with their size. Here is a summary:

• 0 -10 0.00%
• 0 - 100 56.0%
• 0 - 1000 50.4%
• 0 - 10000 31.0%
• 0 - 100000 16.2%
• 0 - 1000000 6.99%

Once a number has more than nine digits, it cannot be a 2D number because at least one digit would then repeat. The upper limit must be below 987,654,320, a number that has an additive digital root of 8 and is thus not a 2D number. I excluded 987,654,321 because additive digital roots lie between 1 and 9 and all those digits are taken. The question that must be asked is what is the largest 2D number? It can contain no more than nine digits and one of those must be zero. Testing revealed that:

• no nine digit number containing the digit 9 can be a 2D number
• 876,543,210 qualifies as a 2D number since it has a digital root of 9

So it is that 876,543,210 is the largest 2D number although all of the 8 x 8! = 322,560 (leading zeros not allowed) possible permutations are of course 2D numbers.

In the range between 26500 and 27000, the percentage of 2D numbers is 17.2%. The numbers are (with my diurnal age shown in bold):

26503, 26504, 26508, 26509, 26513, 26514, 26517, 26518, 26530, 26531, 26539, 26540, 26541, 26548, 26549, 26571, 26578, 26580, 26581, 26584, 26587, 26589, 26590, 26593, 26594, 26598, 26703, 26704, 26708, 26715, 26730, 26740, 26748, 26749, 26751, 26758, 26780, 26784, 26785, 26789, 26794, 26798, 26803, 26805, 26807, 26809, 26814, 26815, 26830, 26834, 26839, 26841, 26843, 26845, 26847, 26850, 26851, 26854, 26857, 26859, 26870, 26874, 26875, 26879, 26890, 26893, 26895, 26897, 26904, 26905, 26908, 26935, 26938, 26940, 26945, 26947, 26950, 26953, 26954, 26958, 26974, 26978, 26980, 26983, 26985, 26987

Here is a permalink to the algorithm that I used to generate these numbers. 

An interesting extension is to consider the multiplicative digital root which is the single digit reached when multiplying the digits of the number together (the results can range from 0 to 9). I've concocted the term Doubly Digitally Distinct or 3D for numbers that satisfy the following criteria:

• no repeated digits
• an arithmetic digital root that is different to any of its digits
• a multiplicative digital root that is different to any of its digits and also to the arithmetic digital root

Applying these criteria to the same range of numbers as earlier (26500 to 27000), we find 11.6% of numbers satisfy. These are:

26513, 26514, 26517, 26518, 26531, 26539, 26541, 26548, 26549, 26571, 26578, 26581, 26584, 26587, 26589, 26593, 26594, 26598, 26715, 26748, 26749, 26751, 26758, 26784, 26785, 26789, 26794, 26798, 26814, 26815, 26834, 26839, 26841, 26843, 26845, 26847, 26851, 26854, 26857, 26859, 26874, 26875, 26879, 26893, 26895, 26897, 26935, 26938, 26945, 26947, 26953, 26954, 26958, 26974, 26978, 26983, 26985, 26987

The number 26870 does not qualify as a 3D number because its multiplicative digital root is 0 and that is one of the digits of the number. In fact, any number containing a zero cannot be a 3D number. However, the nearby 26874 and 26875 both qualify as they have additive digital roots of 9 and 1 respectively and multiplicative digital roots of 0. A similar table to that shown above but this time for 3D numbers looks like this.

• 0 -10  0.00%
• 0 - 100 33.0%
• 0 - 1000  26.7%
• 0 - 10000 14.9%
• 0 - 100000 7.61%
• 0 - 1000000 2.78%

Here is a permalink that can be used to generate the statistics in this table. The first 3D number is 23 and it begins a run of three consecutive such numbers: 23, 24 and 25. We see that:

• 23 has an additive digital root of 5 and a multiplicative digital root of 6
• 24 has an additive digital root of 6 and a multiplicative digital root of 8
• 25 has an additive digital root of 7 and a multiplicative digital root of 0

However, the next number 26 has an additive digital root of 8 and a multiplicative digital root of 2 which is one of the digits of the original number. Thus it does not meet the criteria. The question remains as to what is the largest 3D number. It cannot contain more than eight distinct digits. Testing revealed that:

• no eight digit number containing the digit 9 is a 3D number
• 87,654,321 qualifies as a 3D number
• It has a arithmetic digital root of 9
• it has a multiplicative digital root of 0

So 87,654,321 is the largest 3D number although any of the 8! = 40,320 permutations of those digits will also be a 3D number.

ADDENDUM 

October 30th 2020

It occurred to me that it would also be interesting to look at the "complement" of 2D and 3D numbers. The complement of 2D numbers I will define as numbers that have at least one repeated digit and whose arithmetic digital root is one of the digits of the number. The complement of 3D numbers I will define as numbers that have at least one repeated digit and whose arithmetical digital root and multiplicative digital roots are digits of the number.

Here is a permalink to an algorithm that will identify complementary 3D numbers in the range up to 40,000. I have also made an entry in my Bespoken For Sequences. Such numbers comprise 8.01% of the range. Here are the initial members: 0, 100, 118, 181, 188, 200, 299, 300, 400, 500, 600, 700, 800, 811, 818, 881, 899, 900, 909, 929, 989, 990, 992, 998, 1000.

Numbers like 1000 clearly qualify for membership so let's take the less obvious 998. The number has one repeated digit (9) and its arithmetic digital root is 8 while its multiplicative digital root is also 8. Thus it qualifies too. Clearly such complementary 2D and 3D numbers have no upper bound unlike the 2D and 3D numbers themselves.