Нарциссические числа

A narcissistic number (also called an Armstrong number or pluperfect digital invariant) is a number that equals the sum of its own digits, each raised to the power of the number of digits. For example, 153 = 1³ + 5³ + 3³ = 1 + 125 + 27 = 153. These self-referential numbers are rare and fascinating -- in base 10, only 88 of them exist.

Как определить нарциссическое число

To check if a number is narcissistic: count its digits (call this n), raise each digit to the n-th power, and add the results. If the sum equals the original number, it is narcissistic. Here are some examples:

Полный список по количеству цифр

Narcissistic numbers are rare. Here is the complete inventory organized by the number of digits:

In total, there are exactly 88 narcissistic numbers in base 10. The largest has 39 digits. Beyond that, the maximum possible digit-power sum cannot reach the number itself, so no more can exist.

Свойства нарциссических чисел

История и этимология

The name "narcissistic number" was coined by mathematician D. H. Lehmer in reference to numbers that are fixated on themselves, like the mythological Narcissus. They are also called Armstrong numbers after Michael F. Armstrong, who introduced them in a 1969 assignment for computer science students at the University of Rochester. The OEIS (On-Line Encyclopedia of Integer Sequences) lists them as sequence A005188.

Narcissistic Numbers up to 10.000

Click on any narcissistic number to see its complete mathematical analysis.

Большие нарциссические числа

Beyond the small examples, narcissistic numbers grow quickly in size. Here are some notable larger specimens with their digit-power decompositions:

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• The largest narcissistic number is 4,679,307,774 (10 digits)—no 11-digit narcissistic numbers exist, and none beyond. This complete enumeration (only 88 total in base 10) was computationally verified. The upper bound emerges from growth rate analysis: for large n, the maximum digit sum power (9^n) grows too slowly compared to minimum n-digit number (10^(n-1)), making narcissistic numbers impossible beyond n=10.
• The single 6-digit narcissistic number (548834) is isolated—no nearby numbers share the property. This clustering pattern (some digit lengths containing multiple narcissistics, others none) remains unexplained mathematically. Understanding why clustering occurs at certain digit lengths remains an open question.
• Kaprekar numbers (1 and 9) are sometimes confused with narcissistic numbers, though they follow completely different definitions. A Kaprekar number produces itself through splitting and recombining squared digits. The conceptual similarity leads some to study generalizations combining both properties.
• In base 3, narcissistic numbers exist with different frequencies than base 10. The number 153 in decimal equals specific values in other bases but may not be narcissistic there. This base-dependence reveals how narcissistic properties depend on positional representation rather than intrinsic number characteristics.
• The narcissistic number 9,800,817 (7 digits) equals 9⁷+8⁷+0⁷+0⁷+8⁷+1⁷+7⁷ = 4,782,969 + 2,097,152 + 0 + 0 + 2,097,152 + 1 + 823,543 = 9,800,817. Manual verification of large narcissistics is tedious—computers enable easy verification but discovery requires exhaustive search given their rarity.