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In recreational number theory, a narcissistic number〔(''Perfect and PluPerfect Digital Invariants'' ) by Scott Moore〕 (also known as a pluperfect digital invariant (PPDI),〔( PPDI (Armstrong) Numbers ) by Harvey Heinz〕 an Armstrong number〔(Armstrong Numbers ) by Dik T. Winter〕 (after Michael F. Armstrong)〔(Lionel Deimel’s Web Log )〕 or a plus perfect number) 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 = 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'' = ''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 = 1·32 + 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'' = ''d''''k''''m'' + ''d''''k''-1''m'' + ... + ''d''2''m'' + ''d''1''m'', then ''n'' is called a perfect digital invariant or PDI.〔( PDIs ) by Harvey Heinz〕〔 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.'' 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Narcissistic number」の詳細全文を読む スポンサード リンク
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