The record breakers for the 3
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Figure 1: see blog post |
Figure 2 shows the trajectory for 27859 using a logarithmic scale for the vertical axis.
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Figure 2: permalink |
The record breakers for the 3
![]() |
Figure 1: see blog post |
![]() |
Figure 2: permalink |
Figure 1 shows an interesting result via Cliff Pickover.
Sometimes, when looking for properties of the number associated with my diurnal age, I struggle to find a property that interests me. Today was one such occasion and the number was 27857. However, I did notice a "peculiarity" as I call it and it involved both the sum of the number's digits cubed and the number's proper divisors. Firstly, the digits cubed:
This peculiarity is quite rare and in fact up to
I asked Gemini to explain the statement:
Here's what it came up with.
The statement refers to the relationship between an integer partition and a generating function, which is a way of encoding an infinite sequence of numbers as the coefficients of a formal power series. In this context, the "polynomial equation" is more accurately a generating function, which can be represented as a polynomial-like expression.
A partition of a positive integer
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
The statement means that for any given integer, we can construct a polynomial-like expression (a generating function) that "encodes" all of its possible partitions.
The connection is established through the concept of generating functions. The generating function for the number of partitions of an integer
Each term in the product
And so on for all integers
When you multiply these series together, the coefficient of
Let's find the partitions for the number 3. The partitions are:
3
2 + 1
1 + 1 + 1
There are 3 partitions of 3.
Now, let's see how the generating function gives us this result. We look at the expansion of the product:
From the first factor, take
From the first factor, take
From the third factor, take
The coefficient of
Here is another example for the partitions of 4, which we know is 5.
To find the coefficient of
The total number of ways to get
In essence, the "polynomial" or generating function provides a systematic way to count the number of partitions for any integer by looking at the coefficients of the series expansion.
I'm surprised I've not come across this type of sequence before. It has two variants and they are generated iteratively as follows:
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Figure 1: permalink |
If the sum of prime factors with multiplicity is considered then we get OEIS A121360 with sequence members 1, 8, 14, 26, 62, 134, 393, 1257, 4659, 9314, 27933 up to 100,000. The trajectories are shown in Figure 2 with length indicating the number of steps or iterations:![]() |
Figure 2: permalink |
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Figure 3: permalink |
FIRST EXAMPLE
The numbers 1 and 4 have the interesting properties that:
Yesterday I turned 27487 days old. This is a prime number of days and this number together with the next three prime numbers form the following sequence:$$27847, 27851, 27883, 27893$$Now let's combine these numbers together to from a continued fraction:
1 27847 + ------------------------- 1 27851 + ---------------- 1 27883 + ------- 27893
1 41 + ---------------- 1 43 + ---------- 1 47 + ---- 53
1 223 + ---------------------------------------- 1 227 + --------------------------------- 1 229 + -------------------------- 1 233 + ------------------- 1 239 + ------------ 1 241 + ----- 251