Narcissistic numbers and their secrets | Tangente

For k = 3, one example is 153, since 1³ + 5³ + 3³ = 153. Readers can check that the same holds for 370, 371 and 407. There are three 4-digit examples and three 5-digit ones.

There is only one 6-digit example (namely 548,834, since 5⁶ + 4⁶ + 8⁶ + 8⁶ + 3⁶ + 4⁶ = 548,834), and no 2-, 12- or 13-digit examples, for instance. The record is held at 11 digits, with eight such numbers. Initially described as Armstrong numbers of the first kind or pluperfect digital invariants (also known in English as PPDIs, for PluPerfect Digit Invariant), they were later dubbed narcissistic numbers by some authors. For others, however, this terminology covers a much broader range of integers: those whose value can be calculated from their own digits (excluding, of course, their representation in the decimal system, which itself presupposes a calculation).

There are only finitely many pluperfect digital invariants:

89, to be exact, and the largest is

115,132,219,018,763,992,565,095,597,973,971,522,401.