# Narcissistic/Plus Perfect/Armstrong

• bahamagreen
In summary, the conversation discusses a function that takes the digits of a natural number and sums their third powers, and the claim that 1/3 of all natural numbers will terminate in 153. The speaker found some cycles and four numbers that terminate themselves or any others. They also mention the possibility of an analytical approach to determine the 1/3 claim, but note that it may be difficult due to the complexity of the problem. They also bring up the Collatz conjecture as an example of a similar problem. The conversation ends with a discussion on the behavior of the function for numbers with more than 4 digits.
bahamagreen
I found reference to a function described as taking the digits of a natural number and summing their third powers, iterating the process. The claim was that 1/3 of all natural numbers will terminate the process in 153.

I did this initiating the process with the first 66 positive integers and found some cycles:

160->217->352->160
250->133->55->250

and I found these four that terminate themselves or any others if they appear (1 is trivial)

153
370
371
407

For the first 66 I got 18 "153"s so .2727... maybe approaching 1/3, but more suspiciously like 1/4...

I looked around OEIS and saw this process as the base 10 third power version of the Narcissistic/Plus Perfect/Armstrong numbers... and found these four numbers to be the only ones for "n"=3 where an "n" digit number is the sum of the "n"th powers of its digits.

Is there an analytical approach to determine the 1/3 claim for 153?

bahamagreen said:
For the first 66 I got 18 "153"s so .2727... maybe approaching 1/3, but more suspiciously like 1/4...
The first 66 numbers are not sufficient to see any trend. Try the first 10000?

Every number above 2915 will get smaller in the process, as 9999 -> 2916 -> 954 and numbers below 9999 will get smaller values, of course. Therefore, to find all possible final numbers or final series, it is sufficient to check the first 2915 numbers (probably less with a better analysis).

Is there an analytical approach to determine the 1/3 claim for 153?
First, you need a mathematical definition of "1/3". There is an infinite amount of numbers which terminate in 153, and an infinite amount of numbers which terminate in 370, or any other possible end. Therefore, there are as many numbers ending in 153 as there are numbers ending in 370.
For each n, you can consider the fraction of numbers below ending in 153. That fraction might have a limit, and it could be 1/3. But usually, there is no (known) way to derive that. I would not expect a proof - and even if there is one, I would not expect that it is understandable without a lot of number theory.

For an example of a similar problem, see the Collatz conjecture - the problem is easy to understand, but hard to solve.

"Every number above 2915 will get smaller in the process..."

How did you determine that?

With the analysis after the quoted part.

In general, let f(n) be a function which maps a natural number to its sum of cubed digits.
For any number n with d digits, the maximal f(n) corresponds to the number where all digits are 9. Therefore, f(n)<=9^3*d
At the same time, any number with d digits is at least 10^(d-1) (e.g. any number with 3 digits is at least 100). Therefore, n>=10^(d-1).
Using those equations, is easy to verify that f(n)<n for all numbers with more than 4 digits. For d=4, the analysis can be found in my previous post.

I would approach this claim with skepticism and a desire for further evidence and analysis. While the initial findings may suggest a relationship between the iterative process and the number 153, it is important to consider a larger sample size and to look for patterns or explanations for the cycles and terminating numbers. Additionally, it would be beneficial to explore other values for "n" and see if the 1/3 claim holds true for those as well.

It is also important to note that the concept of Narcissistic/Plus Perfect/Armstrong numbers is a well-studied topic in mathematics, and there may be existing analytical approaches or theories that can be applied to this specific scenario. Further research and collaboration with other experts in the field may provide more insight into the validity of the 1/3 claim for 153.

In conclusion, while the initial findings are intriguing, more evidence and analysis are needed before making a definitive conclusion about the relationship between the iterative process and the number 153. As scientists, it is important to approach claims with an open mind, but also with a critical and analytical perspective.

## 1. What is Narcissistic/Plus Perfect/Armstrong?

Narcissistic/Plus Perfect/Armstrong is a mathematical concept that combines the properties of narcissistic numbers, plus perfect numbers, and Armstrong numbers. It is a special type of number that is equal to the sum of its own digits raised to a power, which makes it a self-referencing number.

## 2. How is a Narcissistic/Plus Perfect/Armstrong number calculated?

To calculate a Narcissistic/Plus Perfect/Armstrong number, each digit of the number is raised to a power and then added together. For example, in the number 153, the digits 1, 5, and 3 would be raised to the power of 3 and then added together to get the result of 153.

## 3. What is the significance of Narcissistic/Plus Perfect/Armstrong numbers?

These numbers have a special significance in mathematics because they are rare and have interesting properties. They are also used in number theory and cryptography.

## 4. How are Narcissistic/Plus Perfect/Armstrong numbers different from each other?

Narcissistic numbers are a special case of Armstrong numbers, where the power is equal to the number of digits. Plus perfect numbers are a subset of Armstrong numbers, where the power is equal to the number of digits minus one. Narcissistic/Plus Perfect/Armstrong numbers combine both of these properties.

## 5. Are there any real-world applications of Narcissistic/Plus Perfect/Armstrong numbers?

While these numbers may not have direct real-world applications, they are used in various mathematical puzzles and problems. They also have applications in coding and cryptography, as well as in understanding the patterns and properties of numbers.

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