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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.

The definition of a narcissistic number relies on the dozenal 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 ≤ E. Such a number n is called narcissistic if it satisfies the condition

n = dkk + dk-1k + ... + d2k + d1k.

For example, the 3-digit dozenal number 577 is a narcissistic number because 577 = 53 + 73 + 73.

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. For example, the dozenal number 11XX has four dozenal digits and is the sum of the cubes (third powers) of its dozenal digits

11XX = 13 + 13 + X3 + X3,

so it is a perfect digital invariant but not a narcissistic number.


There are 73 narcissistic numbers (not include 0), they are:

1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 25, X5, 577, 668, X83, 14765, 938X4, 369862, X2394X, 2380293, 58955X3, EE7332295, 4X24189999, 79003E307X, 4903X15X995, 4X446561XX6, 64508202EX3, 6E32875792X, 72709868E8X, E5X6586E049, E79238X38E3, EX70909E036, EX7584X9XX8, 7E38E60846803, 4029826639111E, 724106687E585E, 7593E3703X8E346, 8315X3E9E7X7652, E463870402XEX8E, 11692939925215X3, 6963788443EE6X87, 71263199E4879E29E43, 26532105935901094X8E, 643896373E63132EE610, 643896373E63132EE611, 91E6EX37E5E2395907X6, 2069E9481507492377719, 2620293539X21867386XE, 2870507436X364X15XE62, 3096XE0892X8209162X8X, 45791X819296E12E27X53, 61035E7E65684375498EX, 645E3539X07EX360880E9, 83771E9E01673XE1XE964, 9E4X42EE88552E80E3368, E73E359723EEE64126E27, 20E446X429978188986395, 225531X5437666X776566E, 3586944EX747E920X39987X4, 260055453E83538E5744348884, 221E3351383X6924E54171565069, 337839406880E85328E93631X8EX, 2577803E895038330E35E32785306X7, 25894E2929727835994701EX53179E6, 267EX8248E5E0616952121220X194X5, 33X6EE19246614126E32X34595E2923, 34X5E53EX3154E85X9768295622EX03, 4E9657XE0E46098142080EEX9437XE0, 4E9657XE0E46098142080EEX9437XE1, 1EX997236136XE0EX788X574814X726560, 1EX997236136XE0EX788X574814X726561, 1253553715X86X851X882041088E9888E35, 173168409X26156653E51X965431E8769E04, 2X5532X6E02105X1921EE353E2X1E6E93E48X10, 2X5532X6E02105X1921EE353E2X1E6E93E48X11, 102434015825787E1510924967E6321002E774440, 102434015825787E1510924967E6321002E774441, 106338439XE23771X82X96E412428X02E64558375, 14419E3894089683E7146110X060E906868452E59, 1460X58E16EE34588139971E557047238X4X12478, 146122093E69EX63X35069652129E5E8X21271044, 2067909536257238618E4E86X65E63EE0X30510E0, 2067909536257238618E4E86X65E63EE0X30510E1, 200E4X6E67E3430064319747100E70E0050XE8X140E, 11775E71382E98173E4314294891E94E6E6265980E59480EX67, 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E

The largest such number (15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E) has 43 digits.

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). 
  • Sum-product numbers (sequence A038369 in the OEIS). 
  • Dudeney numbers (sequence A061209 in the OEIS).
  • Factorions (sequence A014080 in the OEIS).

where di are the digits of n in some base.