Narcissistic number

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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.


The definition of a narcissistic number relies on the decimal 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 ≤ 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.[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 = 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:
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[edit]

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, 22, 122

The sequence of "base 4" narcissistic numbers starts: 0, 1, 2, 3, 130, 131, 203, 223, 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.[1]

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


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]

Related concepts[edit]

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:

where di are the digits of n in some base.


  1. ^ a b c Weisstein, Eric W., "Narcissistic Number", MathWorld.
  2. ^ a b c Perfect and PluPerfect Digital Invariants by Scott Moore
  3. ^ PPDI (Armstrong) Numbers by Harvey Heinz
  4. ^ Armstrong Numbersl by Dik T. Winter
  5. ^ Lionel Deimel’s Web Log
  6. ^ (sequence A005188 in OEIS)
  7. ^ PDIs by Harvey Heinz
  8. ^ Rose, Colin (2005), Radical Narcissistic Numbers, Journal of Recreational Mathematics, 33(4), pages 250-254.
  9. ^ Pretty wild narcissistic numbers - numbers that pwn by Colin Rose

External links[edit]