# Narcissistic number

In recreational number theory, a **narcissistic number**^{[1]}^{[2]} (also known as a **pluperfect digital invariant** (**PPDI**),^{[3]} an **Armstrong number**^{[4]} (after Michael F. Armstrong)^{[5]} or a **plus perfect number**)^{[6]} 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.

## Definition

The definition of a narcissistic number relies on the decimal representation *n* = *d*_{k}*d*_{k-1}...*d*_{1} of a natural number *n*, i.e.,

*n*=*d*_{k}·10^{k-1}+*d*_{k-1}·10^{k-2}+ ... +*d*_{2}·10 +*d*_{1},

with *k* digits *d*_{i} satisfying 0 ≤ *d*_{i} ≤ 9. Such a number *n* is called narcissistic if it satisfies the condition

*n*=*d*_{k}^{k}+*d*_{k-1}^{k}+ ... +*d*_{2}^{k}+*d*_{1}^{k}.

For example the 3-digit decimal number 153 is a narcissistic number because 153 = 1^{3} + 5^{3} + 3^{3}.

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*=*d*_{k}*b*^{k-1}+*d*_{k-1}*b*^{k-2}+ ... +*d*_{2}*b*+*d*_{1},

where the base-*b* digits *d*_{i} satisfy the condition 0 ≤ *d*_{i} ≤ *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 = 1·3^{2} + 2·3 + 2 , and it satisfies the equation 17 = 1^{3} + 2^{3} + 2^{3}.

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*=*d*_{k}^{m}+*d*_{k-1}^{m}+ ... +*d*_{2}^{m}+*d*_{1}^{m},

then *n* is called a **perfect digital invariant** or **PDI**.^{[7]}^{[2]} For example, the decimal number 4150 has four decimal digits and is the sum of the *fifth* powers of its decimal digits

- 4150 = 4
^{5}+ 1^{5}+ 5^{5}+ 0^{5},

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:*

- .

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

The sequence of base 8 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, ... (sequence A010354 and A010351 in OEIS)

The sequence of base 12 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 25, ᘔ5, 577, 668, ᘔ83, ... (sequence A161949 in the OEIS)

The sequence of base 16 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, 64B, ... (sequence A161953 in the OEIS)

The sequence of base 3 narcissistic numbers starts: 0, 1, 2, 12, 22, 122

The sequence of base 4 narcissistic numbers starts: 0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303 (sequence A010344 and A010343 in OEIS)

In base 2, the only narcissistic numbers are 0 and 1.

The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the *k*th 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. Setting *b* equal to 10 shows that the largest narcissistic number in base 10 must be less than 10^{60}.^{[1]}

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.^{[1]}

Clearly, in all bases, all one-digit numbers are narcissistic numbers.

A base *b* has at least one two-digit narcissistic number if and only if *b*^{2} + 1 is not prime, and the number of two-digit narcissistic numbers in base *b* equals , where is the number of positive divisors of *n*.

Every base *b* ≥ 3 that is not a multiple of nine has at least one three-digit narcissistic number. The bases that do not are

- 2, 72, 90, 108, 153, 270, 423, 450, 531, 558, 630, 648, 738, 1044, 1098, 1125, 1224, 1242, 1287, 1440, 1503, 1566, 1611, 1620, 1800, 1935, ... (sequence A248970 in the OEIS)

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]}

## 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 : 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)
^{[8]} - Sum-product numbers (sequence A038369 in the OEIS) :
- Dudeney numbers (sequence A061209 in the OEIS) :
- Factorions (sequence A014080 in the OEIS) :

where *d*_{i} are the digits of *n* in some base.

## References

- 1 2 3 Weisstein, Eric W. "Narcissistic Number".
*MathWorld*. - 1 2 3
*Perfect and PluPerfect Digital Invariants*by Scott Moore - ↑ PPDI (Armstrong) Numbers by Harvey Heinz
- ↑ Armstrong Numbers by Dik T. Winter
- ↑ Lionel Deimel’s Web Log
- ↑ (sequence A005188 in the OEIS)
- ↑ PDIs by Harvey Heinz
- ↑ Rose, Colin (2005), Radical Narcissistic Numbers, Journal of Recreational Mathematics, 33(4), pages 250-254.

- Joseph S. Madachy,
*Mathematics on Vacation*, Thomas Nelson & Sons Ltd. 1966, pages 163-175. - Rose, Colin (2005),
*Radical narcissistic numbers*, Journal of Recreational Mathematics, 33(4), 2004-2005, pages 250-254. -
*Perfect Digital Invariants*by Walter Schneider

## External links

- Digital Invariants
- Armstrong Numbers
- Armstrong Numbers in base 2 to 16
- Armstrong numbers between 1-999 calculator
- Symonds, Ria. "153 ♥ Narcissistic Number".
*Numberphile*. Brady Haran.