Narcissistic number

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In number theory, a narcissistic number^[1]^[2] or pluperfect digital invariant (PPDI)^[3] or Armstrong number^[4] is a number that in a given base is the sum of its own digits to the power of the number of digits.

To put it algebraically, let $n = \sum_{i = 1}^k d_ib^{i - 1}$ be an integer with representation $d k d k - 1 ... d 1$ in base-b notation. If $n = \sum_{i = 1}^k {d_i}^k$ then n is a narcisstic number. For example, the decimal (Base 10) number 153 has three digits and is a narcissistic number, because:

$1^3 + 5^3 + 3^3 = 153\, .$

If the constraint that the power must equal the number of digits is dropped, so that for some m it happens that $n = \sum_{i = 1}^k {d_i}^m$ then n is called a perfect digital invariant or PDI.^[5]^[2] For example, the decimal number 4150 has four digits and is the sum of the fifth powers of its digits

$4^5+1^5+5^5+0^5 = 4150\, ,$

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.

1 Narcissistic numbers in various bases
2 Related concepts
3 References
4 External links

[edit] 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 narcisstic 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.

There are 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.^[1]

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.^[2]

[edit] 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: