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# 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.

The definition of a narcissistic number relies on the decimal representation *n* = *d*_{k}*d*_{k-1}...*d*_{1}*d*_{0} of a natural number *n*, e.g.

*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:

\( 153=1^3+5^3+3^3 \)

\( 370=3^3+7^3+0^3 \)

\( 371=3^3+7^3+1^3 \)

\( 407=4^3+0^3+7^3 \).

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

4 External links

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.[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]

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 : \( n=m^{d_k} + m^{d_{k-1}} + \dots + m^{d_2} + m^{d_1} \) 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

External links

Digital Invariants

Armstrong Numbers

Armstrong numbers between 1-999 calculator

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