Multiplicative and Additive Digital Roots

Even though I've written about multiplicative and arithmetic digital roots in numerous posts, it would seem that I've never addressed the obvious question of how many numbers have identical roots. I was searching for properties of the number associated with my diurnal age (28131) when I noticed the following:$$ \begin{align} 28131 &\rightarrow 2 + 8 + 1+3+1 = 15 \rightarrow 1 + 5 =6 \\ 28131 &\rightarrow 2 \times 8 \times 1 \times 3 \times 1 =48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2 = 6 \end{align}$$It turns out that there are \( \textbf{1085} \) such numbers in the range between 1 and 40000, representing 2.7125% of the range. I won't list all of the numbers here but only those from my diurnal age up to 40000 (permalink):

28131, 28167, 28169, 28176, 28178, 28187, 28196, 28223, 28232, 28311, 28322, 28347, 28374, 28437, 28473, 28617, 28619, 28671, 28691, 28716, 28718, 28734, 28743, 28761, 28781, 28817, 28871, 28916, 28961, 29117, 29126, 29162, 29168, 29171, 29186, 29216, 29261, 29612, 29618, 29621, 29681, 29711, 29816, 29861, 29999, 31113, 31128, 31131, 31139, 31169, 31182, 31193, 31196, 31218, 31227, 31234, 31243, 31272, 31281, 31311, 31319, 31324, 31342, 31344, 31391, 31423, 31432, 31434, 31443, 31619, 31677, 31691, 31722, 31767, 31776, 31778, 31787, 31812, 31821, 31877, 31889, 31898, 31913, 31916, 31931, 31961, 31988, 32118, 32127, 32134, 32143, 32172, 32181, 32217, 32226, 32228, 32262, 32271, 32282, 32314, 32336, 32341, 32363, 32413, 32431, 32478, 32487, 32622, 32633, 32712, 32721, 32748, 32784, 32811, 32822, 32847, 32874, 33111, 33119, 33124, 33142, 33144, 33191, 33214, 33236, 33241, 33263, 33326, 33344, 33362, 33412, 33414, 33421, 33434, 33441, 33443, 33477, 33479, 33497, 33557, 33575, 33623, 33632, 33666, 33747, 33749, 33755, 33774, 33794, 33911, 33947, 33974, 34123, 34132, 34134, 34143, 34213, 34231, 34278, 34287, 34312, 34314, 34321, 34334, 34341, 34343, 34377, 34379, 34397, 34413, 34431, 34433, 34728, 34737, 34739, 34773, 34782, 34793, 34827, 34872, 34937, 34973, 35357, 35375, 35537, 35573, 35735, 35753, 36119, 36177, 36191, 36222, 36233, 36323, 36332, 36366, 36636, 36663, 36717, 36771, 36911, 37122, 37167, 37176, 37178, 37187, 37212, 37221, 37248, 37284, 37347, 37349, 37355, 37374, 37394, 37428, 37437, 37439, 37473, 37482, 37493, 37535, 37553, 37617, 37671, 37716, 37718, 37734, 37743, 37761, 37781, 37817, 37824, 37842, 37871, 37934, 37943, 38112, 38121, 38177, 38189, 38198, 38211, 38222, 38247, 38274, 38427, 38472, 38717, 38724, 38742, 38771, 38819, 38891, 38918, 38981, 39113, 39116, 39131, 39161, 39188, 39311, 39347, 39374, 39437, 39473, 39611, 39734, 39743, 39818, 39881

All permutations of any of these numbers will have multiplicative and arithmetic digital roots that are the same. Putting the digits of 28131 in ascending order, we get 11238. If we only consider numbers whose digits are in ascending order, then in the range up to 40000 there are only \( \textbf{74}\) numbers that qualify. These are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 137, 139, 168, 179, 188, 233, 267, 299, 346, 389, 899, 1124, 1157, 1347, 1355, 1469, 1779, 1788, 2236, 2346, 2348, 2778, 3335, 3779, 11126, 11133, 11148, 11177, 11222, 11238, 11279, 11339, 11369, 11579, 11666, 11677, 11679, 11699, 11999, 12237, 12269, 12334, 12444, 12446, 12678, 12689, 12777, 12788, 13344, 13677, 13778, 13889, 14777, 22236, 22238, 23336, 23478, 29999, 33344, 33477, 33479, 33557, 33666

Permutations of the digits of these numbers will generate the other 1011 (1085 - 74) numbers in the range. These numbers are members of OEIS A064702.

What's Special About 27527?

I've written before about primes whose digits are all prime in two earlier posts:

• 27253: A Very Special Prime on the 15th November 2023
• Prime Digits on 24th October 2023

It's time to revisit this topic because, as I count my diurnal age, I have 27527 coming up in four days time and it is such a prime. These primes are rare indeed and need to be celebrated. Here is the list of the 124 such primes up to 40000 (permalink):

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557, 5573, 5737, 7237, 7253, 7333, 7523, 7537, 7573, 7577, 7723, 7727, 7753, 7757, 22273, 22277, 22573, 22727, 22777, 23227, 23327, 23333, 23357, 23537, 23557, 23753, 23773, 25237, 25253, 25357, 25373, 25523, 25537, 25577, 25733, 27253, 27277, 27337, 27527, 27733, 27737, 27773, 32233, 32237, 32257, 32323, 32327, 32353, 32377, 32533, 32537, 32573, 33223, 33353, 33377, 33533, 33577, 33757, 33773, 35227, 35257, 35323, 35327, 35353, 35527, 35533, 35537, 35573, 35753, 37223, 37253, 37273, 37277, 37337, 37357, 37537, 37573

As can be seen, it's been a while since the last such prime (27337) and it will be quite a while to the next (27733). Furthermore, 27527 has the added property that the sum of its digits (23) is also prime. There are only 61 of these numbers in the range up to 40000. They are (permalink):

2, 3, 5, 7, 23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3527, 3727, 5233, 5237, 5273, 5323, 5527, 7237, 7253, 7523, 7723, 7727, 22573, 23327, 25237, 25253, 25523, 27253, 27527, 32233, 32237, 32257, 32323, 32327, 33223, 33353, 33377, 33533, 33773, 35227, 35353, 35533, 35537, 35573, 35753, 37223, 37337

Figure 1 shows the distribution of these primes:

Figure 1: permalink

Not of all of these prime sums of digits have digits that are all prime. 27527 with its digit sum of 23 qualifies but others don't. In fact there are only 15 numbers with digit sums that are prime and that have all of the digits of this sum prime. These are (permalink):

2, 3, 5, 7, 23, 223, 2777, 7727, 27527, 33377, 33773, 35537, 35573, 35753, 37337

Similarly, not all of these previously listed primes have a digital arithmetic root that is prime. However, 27527 does because its sum of digits (23) leads to a root of 5. There are only 19 such primes in the range up to 40000. They are (permalink):

2, 3, 5, 7, 23, 223, 227, 353, 2333, 2777, 3323, 7727, 27527, 33377, 33773, 35537, 35573, 35753, 37337

So in summary 27527 is special because:

• it is prime: 27527
• all its digits are prime: 2, 5 and 7
• the sum of its digits is prime: 23
• the digits of its sum of digits are prime: 2 and 3
• its arithmetic digital root is prime: 5

All these properties fall within the realm of recreational Mathematics because they are base dependent but nonetheless they are interesting to numberphiles who find number properties of any type in any base to be fascinating.