- Narcissistic number
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. This definition depends on the base b of the number system used, e.g. b = 10 for the decimal system or b = 2 for the binary system.
The definition of a narcissistic number relies on the decimal representation n = dkdk-1...d1d0 of a natural number n, e.g.
- n = dk·10k-1 + dk-1·10k-2 + ... + d2·10 + d1,
with k digits di satisfying 0 ≤ di ≤ 9. Such a number n is called narcissistic if it satisfies the condition
- n = dkk + dk-1k + ... + d2k + d1k.
For example the 3-digit decimal number 153 is a narcissistic number because 153 = 13 + 53 + 33.
Narcissistic numbers can also be defined with respect to numeral systems with a base b other than b = 10. The base-b representation of a natural number n is defined by
- n = dkbk-1 + dk-1bk-2 + ... + d2b + d1,
where the base-b digits di satisfy the condition 0 ≤ di ≤ b-1. For example the (decimal) number 17 is a narcissistic number with respect to the numeral system with base b = 3. Its three base-3 digits are 122, because 17 = 1·32 + 2·3 + 2 , and it satisfies the equation 17 = 13 + 23 + 23.
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,
- 4150 = 45 + 15 + 55 + 05,
so it is a perfect digital invariant but not a narcissistic number.
- There are just four numbers, after unity, which are the sums of the cubes of their digits:
- 153 = 13 + 53 + 33
- 370 = 33 + 73 + 03
- 371 = 33 + 73 + 13
- 407 = 43 + 03 + 73.
- These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.
Narcissistic numbers in various bases
The sequence of "base 4" narcissistic numbers starts: 0, 1, 2, 3, 313
The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the kth powers of a k digit number in base b is
and if k is large enough then
in which case no base b narcissistic number can have k or more digits.
There are 88 narcissistic numbers in base 10, of which the largest is
with 39 digits.
Unlike narcissistic numbers, no upper bound can be determined for the size of PDIs in a given base, and it is not currently known whether or not the number of PDIs for an arbitrary base is finite or infinite.
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 (sequence A046253 in OEIS) :
- Ascending power numbers (sequence A032799 in OEIS) :
- Friedman numbers (sequence A036057 in OEIS).
- Sum-product numbers (sequence A038369 in OEIS) :
- Dudeney numbers (sequence A061209 in OEIS) :
- Factorions (sequence A014080 in OEIS) :
where di are the digits of n in some base.
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