More Numbers Within Numbers

In December of 2022, I made a post titled Numbers Within Numbers and so for this post I've made the title More Numbers Within Numbers but the types of numbers considered in the former post are quite different to the ones I'll be considering in this post. The idea for this post came from a peculiarity in the number associated with my diurnal age today: 27567.

What I mean by a number within a number in this present context is simply a substring of the number viewed as a string. Here the substring being considered is "27" that is contained within the larger string "27567". Thus we see that:$$\mathbf{\text{27}}567$$Now let's consider the sum of digits of the number:$$\mathbf{\text{27}}567\to 2+7+5+6+7=\mathbf{\text{27}}$$Let's move on to the factorisation where we have:$$\mathbf{\text{27}}567=\mathbf{\text{27}}\times 1021$$Now what about the totient? We find that 27567 has a totient of 18360 and$$18360=5\times 8\times 17\times \mathbf{\text{27}}$$Lastly, let's find the absolute value of the determinant of the circulant matrix of 27567. It turns out to be 6777 and$$6777=\mathbf{\text{27}}\times 251$$Thus in the case of 27567, the number within a number (27) turns up:

• in the digits of the number
• as the sum of the digits of the number
• in the factors of the number
• in the factors of the totient
• in the factors of the determinant of the circulant matrix

It can be noted that 27567 is a Harshad number since it is a multiple of its sum of digits (27), and also a Moran number because the ratio is a prime number: 1021 = 27567 / (2 + 7 + 5 + 6 + 7).

The natural question to ask then is how many numbers in the range up to 40000 have this property? It turns out that there are only 18 such numbers with details as shown in Table 1.

Table 1: permalink

So what about other numbers? Let's start with a substring "1". There are five numbers satisfying the previous criteria in the range up to 40000. See Table 2.

Table 2: permalink

For substring "2", there are four numbers satisfying the criteria. See Table 3.

Table 3: permalink

For the substring "3", there are no numbers that meet the criteria. For the substring "4", there are appropriately four numbers that satisfy. See Table 4.

Table 4: permalink

For the substring "5", there are three numbers that meet the criteria. See Table 5.

Table 5: permalink

For substrings "6" and "7", no numbers qualify but for the substring "8" there are three numbers that do. See Table 6.

Table 6: permalink

There are no numbers that satisfy for "9" but there are 23 numbers that satisfy for "10". See Table 7.

Table 7: permalink

For "11", no numbers that satisfy but for "12" there are 39 numbers that satisfy. See Table 8.

Table 8: permalink

For "13", there are two numbers that satisfy. See Table 9.

Table 9: permalink

For "14", there are three numbers that satisfy. See Table 10.

Table 10: permalink

For "15", there are 26 numbers that satisfy. See Table 11.

Table 11: permalink

For "16", there are 14 numbers that satisfy. See Table 12.

Table 12: permalink

For "17", there is only one number. See Table 13.

Table 13: permalink

For "18", there is a grand total of 69 numbers that qualify. See Table 14.

Table 14: permalink

For "19", there are no numbers that qualify but for "20" there are four. See Table 15.

Table 15: permalink

For "21", there are three numbers that qualify. See Table 16.

Table 16: permalink

For "22", there are two numbers that satisfy. See Table 17.

Table 17: permalink

For "23", there is only one number that satisfies. See Table 18.

Table 18: permalink

For "24", there are nine numbers that satisfy. See Table 19.

Table 19: permalink

For "25", there are four numbers that satisfy. See Table 20.

Table 20: permalink

For "26", there are no numbers that qualify and we've already dealt with "27". For "28", there is only one number that satisfies. See Table 21.

Table 21: permalink

That's it for particles in the range up to 40,000 as the SOD criterion makes it difficult  for the digit sum to reach these higher particles. Just to illustrate with an example. Take the particle "29". If we extend the range to one million, then instead of the zero for the range up to 40,000, we have nine numbers that satisfy the criteria. See Table 21.

Table 21: permalink

So, an interesting exercise but purely confined to the realm of recreational Mathematics. There no real reason to conflate the digit sum of a number with its factors as well as its totient and the determinant of its circulant matrix.