Published March 15, 2026
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PERFECT R-NARCISSISTIC NUMBERS IN ANY BASE
Description
We defined in [3] the perfect r-narcissistic numbers, or rppdi, which are a natural extension of the classic ppdi or Armstrong number of the first kind ([1]). We have shown that the set of these rppdi is finite and we have given the list of the first 15 rppdi greater than 1 in decimal base ([4], [OEIS2]).Let us now consider the ppdi and the rppdi in any base b; we will thus define two distinct families (ppdib, ppdi0) on the one hand and (rppdib, rppdi0) on the other hand, associated with any base b other than the decimal base.We will treat here the cases of bases from 3 to 9 (base 2 only has the trivial solution 1 and base 10 corresponds to the classic ppdi and rppdi) without taking into account the common trivial solution 1.
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Additional details
Dates
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2026-03-15
References
- [1] Armstrong numbers of the first kind or ppdi, OEIS A005188 .[2] O.Cira, F.Smarandache, Various arithmetic functions and their applications, PONS asbl, Bruxelles, 2016.[3] R.L.Clerc, Les transformations agréables et une nouvelle classe de nombres narcissiques parfaits, p.1-17, (zenodo.org/records/19004709, 2026), 2022.[4] R.L.Clerc, The perfect r-narcissistic numbers, p.1-2, ( zenodo.org/records/19030196, 2026), 2023.[5] R.L.Clerc, Quelques nombres de Niven-Harshad particuliers, p.1-18, (zenodo.org/records/19018760, 2026), 2023.[OEIS1] Armstrong numbers of the first kind in base 9, OEIS A010353 , N.J.A. Sloane, J.Myers; with references to bases 4 to 16, 2009.[OEIS2] R-narcissistic numbers (rppdi), OEIS A364601 , R.L.Clerc, 2023.