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:27567Now let's consider the sum of digits of the number:27567→2+7+5+6+7=27Let's move on to the factorisation where we have:27567=27×1021Now what about the totient? We find that 27567 has a totient of 18360 and18360=5×8×17×27Lastly, let's find the absolute value of the determinant of the circulant matrix of 27567. It turns out to be 6777 and6777=27×251Thus 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.
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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.
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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.
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Table 4: permalink |
For the substring "5", there are three numbers that meet the criteria. See Table 5.
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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.
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Table 6: permalink |
There are no numbers that satisfy for "9" but there are 23 numbers that satisfy for "10". See Table 7.
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Table 7: permalink |
For "11", no numbers that satisfy but for "12" there are 39 numbers that satisfy. See Table 8.
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Table 10: permalink |
For "15", there are 26 numbers that satisfy. See Table 11.
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Table 12: permalink |
For "17", there is only one number. See Table 13.
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Table 14: permalink |
For "19", there are no numbers that qualify but for "20" there are four. See Table 15.
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Table 15: permalink |
For "21", there are three numbers that qualify. See Table 16.
For "25", there are four numbers that satisfy. See Table 20.
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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.
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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.
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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.
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