ABSTRACT-A brief definition of perfect r-narcissistic numbers (or rppdi), a natural extension of the classic Armstrong numbers (or perfect narcissistic numbers or ppdi), a demonstration of their finiteness and the list of 15 rppdi greater than 1 currently calculated.
- DEFINITION.
Given a k-digit number m (k > 0), we note:
m = ∑i=1i=k ai 10k-i,
Sp(m) = ∑i=1i=k aip, p >= 1.
A k-digit number m is perfect r-narcissistic (or rppdi) if there exists an integer r > 1 such that the sum of the powers k of the digits of mr is equal to m.
The case r = 1 corresponds to the classic perfect narcissistic numbers or Armstrong numbers, or ppdi ([1]); we know that there are 88 such numbers.
An rppdi is still a k-digit fixed point of the Skr transformation ([2])
m ---> Skr(m) = Sk(mr)
We showed in [2] that these Skr transformations are pleasant (cf. Remark 1 below), that is to say they are everywhere convergent and have a finite number of attractors and in particular a finite number of fixed points with k digits (let us observe that Skr has a finite number of fixed points, some of which have exactly k digits, cf. Remark 2 below). There is therefore a finite number of rppdi, each of these numbers being defined by a unique r greater than 1; however, there can theoretically be r-narcissistic numbers for several different values of r (but we can reasonably conjecture that this is not the case).
For the 15 calculated rppdi (numbers from 1 to 10 digits) greater than 1, the corresponding r (> 1) is unique; only the trivial solution 1 is r-narcissistic for all r.
For example, note 7(4) to indicate that 7 is 4-narcissistic.
- PROPERTY.
There are a finite number of perfect r-narcissistic numbers.
The 15 smallest perfect r-narcissistic numbers (r >= 2) greater than 1 are:
7(4), 8(3), 9(2),
180(6), 205(2),
38998(2), 45994(2), 89080(2),
726191(2),
5540343(3), 7491889(2), 8690141(3),
167535050(3), 749387107(4).
9945245922(3).
-REMARK 1: demonstration of Skr pleasant transformation for any (k, r).
Any p-digit number being strictly less than 10p, it comes
Skr(m) <= rp9k
For any pair (k, r) of strictly positive and finite integers, there exists an upper bound
M = r9k such that for any integer m with p digits we have the increase
Skr(m) <= pM.
Consequently, as soon as the number pM has strictly less than p digits, the transformation Skr cannot have a fixed point at p or more than p digits (since we cannot ensure Skr(f) = f ); nor can it then have a cycle with elements with p or more than p digits (iteration by Skr would not make it possible to find an element of strictly more than p digits).
This is ensured as soon as p satisfies the inequality 10p-1/p > M and there obviously always exists a smallest p having this property for any pair (k, r).
We will note p* = p*(k, r) the smallest p such that 10p-1/p > M.
The function 10p-1/p being increasing, for all p > = p*, we always have p M < 10p-1 and the iteration by S< sub>kr of a number with such p digits always provides a number with at most p-1 digits (each iteration causes the transform to lose at least one digit).
Let us observe that for all k and all r strictly positive, we will always have p* >= 3.
It follows that all the attractors of Skr are necessarily made up of numbers contained in the finite interval [0, 10p*-1[, since this p* is still the smallest p such that the upper bound p M is strictly less than 10p-1 (i.e. again, has at most (p-1) digits).
The transformations Skr are everywhere convergent in N towards a finite number of attractors and are therefore all pleasant transformations, the convergence interval being [0, 10< sup>p*-1[; any number it contains has all its iterations in it and any number outside it ends up there after a finite number of iterations.
It follows that it is enough to seek the convergence of all integers strictly less than 10p*-1 to determine all the attractors of Skr.
The pleasant nature of Skr transformations implies in particular that there are only a finite number of rppdi.
- REMARK 2: examples of fixed points of Skr.
We will observe that if the transformations Skr, k >= 1, r >= 2, have many fixed points with strictly more than k digits, they have quite few with exactly k digits ...
S36 has the 6-narcissistic number 180 but also the fixed point 3172;
S52 has the three 2-narcissistic numbers 38998, 45994, 89080 but also the fixed points 128230 and 244353;
S53 no 5-digit 3-narcissistic number, but the fixed points 137023, 163451, 192412, 233722;
S34 no 3-digit 4-narcissistic number, but the fixed point 2110;
S24 no 2-digit 4-narcissistic number, but fixed points 130 and 235... (*) Professeur honoraire Université Paul Sabatier, Toulouse, France, Webmaster du site SAYRAC .
This article was published in HAL Open Archive on 05/10/2023 ( hal-04229895 ). REFERENCES
[1] Armstrong numbers, OEIS A005188 .
[2] R.L.Clerc, Les transformations agréables et une nouvelle classe de nombres narcissiques parfaits, p.1-17, ( https://hal.archives-ouvertes.fr/hal-03619147), 2022.
