Digital Roots and Additive Persistence

To quote from my recent post titled SOD ET AL (Sum Of Digits And Other Things) on June 29th 2021:

DIGITAL ROOT 

While I've not made a specific post about digital roots, I've nonetheless mentioned them in the following posts:

• Heptagonal Numbers
• Star Numbers
• The Goldbach Conjecture and Lucky Numbers
• An Unhappy Family

To quote from Wikipedia:

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0).

Associated with the digital root is the concept of additive persistence defined as:

The additive persistence counts how many times we must sum its digits to arrive at its digital root. For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9. 

Recently I had cause to visit digital roots again in the context of a new sequence that I devised in response to finding a meaningful OEIS sequence associated with the number 26396 which factorises to 2 * 2 * 6599 and that has prime factors of 2 and 6599. I noticed that both of these prime factors have a digital root of 2. This gave me the idea for the following sequence:

S003: Numbers with more than one prime factor such that the digital root of all prime factors is the same.

I developed an algorithm (permalink) to determine all such numbers up to 30,000 and it turned out that there are 1658 numbers in that range, constituting 6.29%. The members of the sequence below 1000 are:

22, 44, 58, 88, 94, 115, 116, 166, 176, 188, 202, 205, 232, 242, 274, 295, 301, 319, 332, 346, 352, 376, 382, 403, 404, 427, 454, 464, 484, 517, 526, 548, 553, 562, 565, 575, 634, 638, 655, 664, 679, 692, 703, 704, 706, 745, 752, 764, 778, 808, 835, 871, 886, 901, 908, 913, 922, 928, 943, 958, 968

It was a short step then to my next sequence where membership is a little more exclusive:

S004: Numbers with more than one prime factor such that the digital root of all prime factors is the same as the digital root of the number itself.

Here is the permalink to the algorithm that I developed. It turns out that there are 106 numbers in the range up to 30,000 and these constitute 0.402 % of the range. Below 1000, there are only two numbers that satisfy this criterion and interestingly they form a pair.

703 = 19 * 37 where 19, 37 and 703 all have a digital root of 1 and 704 = 2^6 * 11 where 2, 11 and 704 all have a digital root of 2. In the range up to 30,000, there are two other pairs:

14527 and 14528

29503 and 29504

The 106 members of this sequence in the range up to 30,000 are:

[703, 704, 1387, 1856, 2071, 2413, 2701, 3008, 3097, 3439, 3781, 3872, 4033, 4699, 5149, 5312, 5833, 6031, 6464, 6697, 7201, 7363, 7543, 7957, 8227, 8768, 9253, 9271, 9937, 10027, 10208, 10279, 10963, 11072, 11359, 11647, 11899, 11989, 12224, 13213, 13357, 13843, 14023, 14041, 14383, 14527, 14528, 14689, 14749, 15317, 15409, 15751, 16021, 16544, 16777, 16832, 17461, 17767, 17803, 17984, 18019, 18829, 19171, 19351, 19729, 19783, 20017, 20197, 20288, 20519, 20701, 20923, 21223, 21296, 21349, 21691, 21907, 22249, 22411, 22592, 22681, 22987, 23347, 24301, 24643, 24896, 25273, 25721, 26011, 26353, 26912, 27037, 27097, 27343, 27667, 27721, 28009, 28352, 28981, 29089, 29216, 29431, 29503, 29504, 29539, 29773]

 Figure 1 shows a plot of these numbers:

Figure 1

I'll make a note about additive persistence. The distribution from 0 to 30,000 is:

• 10 numbers have a digital persistence of 0 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• 1531 have a digital persistence of 1
• 25292 have a digital persistence of 2
• 3168 have a digital persistence of 3

The smallest numbers to have persistences of 0, 1, 2 and 3 are 0, 10, 19 and 199 respectively. The first number with an additive persistence of 4 is 19999999999999999999999. Beyond that, the numbers are ridiculously large. Figure 2 shows the relative proportions:

Figure 2