In recreational number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) is a number that is the sum of its own digits each raised to the power of the number of digits.
The definition of a narcissistic number relies on the dozenal representation n = dkdk-1...d1 of a natural number n, i.e.,
- n = dk·10k-1 + dk-1·10k-2 + ... + d2·10 + d1,
with k digits di satisfying 0 ≤ di ≤ E. Such a number n is called narcissistic if it satisfies the condition
- n = dkk + dk-1k + ... + d2k + d1k.
For example, the 3-digit dozenal number 577 is a narcissistic number because 577 = 53 + 73 + 73.
If the constraint that the power must equal the number of digits is dropped, so that for some m possibly different from k it happens that
- n = dkm + dk-1m + ... + d2m + d1m,
then n is called a perfect digital invariant or PDI. For example, the dozenal number 11XX has four dozenal digits and is the sum of the cubes (third powers) of its dozenal digits
- 11XX = 13 + 13 + X3 + X3,
so it is a perfect digital invariant but not a narcissistic number.
There are 73 narcissistic numbers (not include 0), the first few of them are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 25, X5, 577, 668, X83, 14765, 938X4, 369862, X2394X, and the largest of which is 43-digit 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E
The term "narcissistic number" is sometimes used in a wider sense to mean a number that is equal to any mathematical manipulation of its own digits. With this wider definition narcisstic numbers include:
- Constant base numbers : for some m.
- Perfect digit-to-digit invariants or Münchhausen numbers (sequence A046253 in the OEIS).
- Ascending power numbers (sequence A032799 in the OEIS).
- Friedman numbers (sequence A036057 in the OEIS).
- Radical narcissistic numbers (sequence A119710 in the OEIS).
- Sum-product numbers (sequence A038369 in the OEIS).
- Dudeney numbers (sequence A061209 in the OEIS).
- Factorions (sequence A014080 in the OEIS).
where di are the digits of n in some base.