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Narcissistic number

In 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] 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.

DefinitionEdit

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 122 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  , and forms a cycle of period  . A narcissistic number is a sociable narcissistic number with  , and a 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  .[1]

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, ... (sequence A248970 in the OEIS)

There are only 89 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]

Narcissistic numbers and cycles of Fb for specific bEdit

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   A010344 and A010343
5 0, 1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, ... 18

1234 → 2404 → 4103 → 2323 → 1234

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

1044302 → 2110314 → 1044302

1043300 → 1131014 → 1043300

A010346
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

A010348
7 0, 1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, ... 60 A010350
8 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, ... 63 A010354 and A010351
9 0, 1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, ... 59 A010353
10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, ... 89 A005188
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 A0161948
12 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 25, A5, 577, 668, A83, 14765, 938A4, 369862, A2394A, ... 88 A161949
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 A0161950
14 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 136, 409, 74AB5, 153A632, ... 103 A0161951
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 A0161952
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, ... 294 A161953

Extension to negative integersEdit

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

Programming exampleEdit

The example below implements the narcissistic function described in the definition above to search for narcissistic functions and cycles in Python.

def ppdif(x, b):
    """Pluperfect digital invariant."""
    y = x
    digit_count = 0
    while y > 0:
        digit_count = digit_count + 1
        y = y // b
    total = 0
    while x > 0:
        total = total + pow(x % b, digit_count)
        x = x // b
    return total

def ppdif_cycle(x, b):
    seen = []
    while x not in seen:
        seen.append(x)
        x = ppdif(x, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = ppdif(x, b)
    return cycle

See alsoEdit

ReferencesEdit

  1. ^ a b c Weisstein, Eric W. "Narcissistic Number". MathWorld.
  2. ^ Perfect and PluPerfect Digital Invariants Archived 2007-10-10 at the Wayback Machine by Scott Moore
  3. ^ PPDI (Armstrong) Numbers by Harvey Heinz
  4. ^ Armstrong Numbers by Dik T. Winter
  5. ^ Lionel Deimel’s Web Log
  6. ^ (sequence A005188 in the OEIS)
  • 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 linksEdit