Perfect Digital Invariants

Introduction

Armstrong or narcissistic numbers, Perfect digital invariants (PDI) and related numbers are those numbers which are equal to a sum of certain powers of their digits in some base,

$n=\sum _{i=1}^{k}{d_{i}}^{e_{i}}$   when   $n=\sum _{i=1}^{k}d_{i}\cdot b^{i-1}$   is the base-b expansion of n

(d_k ≠ 0 except for authors who use k = 1 for n = 0).
For Armstrong or narcissistic or plus perfect or pluperfect numbers, all exponents must equal the number k of digits of n in the given base, with b = 10 if not specified otherwise. These sequences are always finite. Perfect digital invariants (PDI) have the more relaxed condition that the exponents can be equal to any fixed number. These sequences can be infinite. Even more general are the so-called Handsome numbers, were each exponent can have any value. Further variants require the exponents to be > 1 or > 0 or even zero, in which case zero digits contribute as 0^0 = 1 unless this is excluded.

Sequences

Type \ Base b:               base 3    base 4    base 5    base 6    base 7    base 8    base 9    base 10  
Armstrong numbers:          (A162216)  A010344   A010346   A010348   A010350   A010354   A010353   A005188.
written in base b:                     A010343   A010345   A010347   A010349   A010351   A010352    

Perfect Digital Invariants:  A162216   A162219   A162222   A162225   A162228   A162231   A162234   A023052  
Handsome numbers: 
 
Base-b Armstrong (b > 10):   A161948 (11)   A161949 (12)   A161950 (13)   A161951 (14)   A161952 (15)   A161953 (16).

Authorship

M. F. Hasler, Perfect Digital Invariants.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Perfect_Digital_Invariants]. Initial version published on Nov. 26, 2019