In number theory, a narcissistic numberPerfect and PluPerfect Digital Invariants by Scott Moore (also known as a pluperfect digital invariant (PPDI), PPDI (Armstrong) Numbers by Harvey Heinz an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base is a number that is the sum of its own digits each raised to the power of the number of digits.

Definition Let be a natural number. We define the narcissistic function for base to be the following:

where is the number of digits in the number in base , and

is the value of each digit of the number. A natural number is a narcissistic number if it is a fixed point for , which occurs if . The natural numbers are trivial narcissistic numbers for all , all other narcissistic numbers are nontrivial narcissistic numbers.

For example, the number 153 in base is a narcissistic number, because and .

A natural number is a sociable narcissistic number if it is a periodic point for , where for a positive integer (here is the th iterate of ), and forms a cycle of period . A narcissistic number is a sociable narcissistic number with , and an amicable narcissistic number is a sociable narcissistic number with .

All natural numbers are preperiodic points for , regardless of the base. This is because for any given digit count , the minimum possible value of is , the maximum possible value of is , and the narcissistic function value is . Thus, any narcissistic number must satisfy the inequality . Multiplying all sides by , we get , or equivalently, . Since , this means that there will be a maximum value where , because of the exponential nature of and the linearity of . Beyond this value , always. Thus, there are a finite number of narcissistic numbers, and any natural number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point. Setting equal to 10 shows that the largest narcissistic number in base 10 must be less than .

The number of iterations needed for to reach a fixed point is the narcissistic function's persistence of , and undefined if it never reaches a fixed point.

A base has at least one two-digit narcissistic number if and only if is not prime, and the number of two-digit narcissistic numbers in base equals , where is the number of positive divisors of .

Every base that is not a multiple of nine has at least one three-digit narcissistic number. The bases that do not are 2, 72, 90, 108, 153, 270, 423, 450, 531, 558, 630, 648, 738, 1044, 1098, 1125, 1224, 1242, 1287, 1440, 1503, 1566, 1611, 1620, 1800, 1935, ...

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.

Narcissistic numbers and cycles of Fb for specific b All numbers are represented in base . '#' is the length of each known finite sequence. Narcissistic numbers # Cycles OEIS sequence(s) 2 0, 1 2 3 0, 1, 2, 12, 22, 122 6 4 0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303 12 and 5 0, 1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, 434434444, ... 18 1234 → 2404 → 4103 → 2323 → 1234

3424 → 4414 → 11034 → 20034 → 20144 → 31311 → 3424

1044302 → 2110314 → 1044302

1043300 → 1131014 → 1043300 6 0, 1, 2, 3, 4, 5, 243, 514, 14340, 14341, 14432, 23520, 23521, 44405, 435152, 5435254, 12222215, 555435035 ... 31 44 → 52 → 45 → 105 → 330 → 130 → 44

13345 → 33244 → 15514 → 53404 → 41024 → 13345

14523 → 32253 → 25003 → 23424 → 14523

2245352 → 3431045 → 2245352

12444435 → 22045351 → 30145020 → 13531231 → 12444435

115531430 → 230104215 → 115531430

225435342 → 235501040 → 225435342 7 0, 1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, 205145, 1424553, 1433554, 3126542, 4355653, 6515652, 125543055, ... 60 8 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, 40663, 42710, 42711, 60007, 62047, 636703, 3352072, 3352272, ... 63 and 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, 2225764, 6275850, 6275851, 12742452, ... 59 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, ... 88 11 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 56, 66, 105, 307, 708, 966, A06, A64, 8009, 11720, 11721, 12470, ... 135 12 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 25, A5, 577, 668, A83, 14765, 938A4, 369862, A2394A, ... 88 13 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, 14, 36, 67, 77, A6, C4, 490, 491, 509, B85, 3964, 22593, 5B350, ... 202 14 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 136, 409, 74AB5, 153A632, ... 103 15 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, 78, 88, C3A, D87, 1774, E819, E829, 7995C, 829BB, A36BC, ... 203 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, 64B, 8C0, 8C1, 99A, AA9, AC3, CA8, E69, EA0, EA1, B8D2, 13579, 2B702, 2B722, 5A07C, 5A47C, C00E0, C00E1, C04E0, C04E1, C60E7, C64E7, C80E0, C80E1, C84E0, C84E1, ... 294

Extension to negative integers Narcissistic numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.

See also Arithmetic dynamics Dudeney number Factorion Happy number Kaprekar's constant Kaprekar number Meertens number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number

References

Joseph S. Madachy, Mathematics on Vacation, Thomas Nelson & Sons Ltd. 1966, pages 163-175. Rose, Colin (2005), Radical narcissistic numbers, Journal of Recreational Mathematics, 33(4), 2004–2005, pages 250-254. Perfect Digital Invariants by Walter Schneider

External links Digital Invariants Armstrong Numbers Armstrong Numbers in base 2 to 16 Armstrong numbers between 1-999 calculator