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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,

then n is called a perfect digital invariant or PDI. For example, the decimal number 4150 has four decimal digits and is the sum of the fifth powers of its decimal digits

4150 = 45 + 15 + 55 + 05,

so it is a perfect digital invariant but not a narcissistic number.

In "A Mathematician's Apology", G. H. Hardy wrote:

There are just four numbers, after unity, which are the sums of the cubes of their digits:

$$153=1^3+5^3+3^3$$
$$370=3^3+7^3+0^3$$
$$371=3^3+7^3+1^3$$
$$407=4^3+0^3+7^3$$.

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.

Contents

1 Narcissistic numbers in various bases
2 Related concepts
3 References

Narcissistic numbers in various bases

The sequence of "base 10" narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474 ... (sequence A005188 in OEIS)

The sequence of "base 3" narcissistic numbers starts: 0, 1, 2, 12, 122

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

$$k(b-1)^k\, ,$$

and if k is large enough then

$$k(b-1)^k<b^{k-1}\, ,$$

in which case no base b narcissistic number can have k or more digits. Setting b equal to 10 shows that the largest narcissistic number in base 10 must be less than 1060.

There are only 88 narcissistic numbers in base 10, of which the largest is

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

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.
Related concepts

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 : $$n=m^{d_k} + m^{d_{k-1}} + \dots + m^{d_2} + m^{d_1}$$ for some m.
Perfect digit-to-digit invariants (sequence A046253 in OEIS) : $$n = d_k^{d_k} + d_{k-1}^{d_{k-1}} + \dots + d_2^{d_2} + d_1^{d_1}\, ,\text{ e.g. } 3435 = 3^3 + 4^4 + 3^3 + 5^5\,$$.
Ascending power numbers (sequence A032799 in OEIS) : n = d_k^1 + d_{k-1}^2 + \dots + d_2^{k-1} + d_1^k\, ,\text{ e.g. } 135 = 1^1 + 3^2 + 5^3 \, .
Friedman numbers (sequence A036057 in OEIS).
Sum-product numbers (sequence A038369 in OEIS) : $$n=\left(\sum_{i=1}^{k}{d_i}\right) \left(\prod_{i=1}^{k}{d_i}\right) \, ,\text{ e.g. } 144 = (1+4+4) \times (1 \times4 \times 4) \,$$.
Dudeney numbers (sequence A061209 in OEIS) : $$n=\left(\sum_{i=1}^{k}{d_i}\right)^3\, ,\text{ e.g. } 512 = (5+1+2)^3 \,$$ .
Factorions (sequence A014080 in OEIS) : $$n=\sum_{i=1}^{k}{d_i}!\, ,\text{ e.g. } 145 = 1! + 4! + 5! \,$$.

where di are the digits of n in some base.
References

^ a b c Weisstein, Eric W., "Narcissistic Number" from MathWorld.
^ a b c Perfect and PluPerfect Digital Invariants by Scott Moore
^ PPDI (Armstrong) Numbers by Harvey Heinz
^ Armstrong Numbersl by Dik T. Winter
^ Lionel Deimel’s Web Log
^ (sequence A005188 in OEIS)
^ PDIs by Harvey Heinz

Joseph S. Madachy, Mathematics on Vacation, Thomas Nelson & Sons Ltd. 1966, pages 163-175.
Perfect Digital Invariants by Walter Schneider
On a curious property of 3435 by Daan van Berkel