Narcissistic number

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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] 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[edit]

The definition of a narcissistic number relies on the decimal 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 1 ≤ dk ≤ 9, and 0 ≤ di ≤ 9 for i < k. Such a number n is called narcissistic if it satisfies the condition

n = dkk + dk-1k + ... + d2k + d1k.

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 = dkbk-1 + dk-1bk-2 + ... + d2b + d1,

where the base-b digits di satisfy the condition 0 ≤ di ≤ 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.

Narcissistic numbers in various bases[edit]

The sequence of base 10 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, ... (sequence A005188 in the OEIS)

The sequence of base 8 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, ... (sequence A010354 and A010351 in OEIS)

The sequence of base 12 narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , Ɛ, 25, ᘔ5, 577, 668, ᘔ83, ... (sequence A161949 in the OEIS)

The sequence of base 16 narcissistic numbers starts: 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, ... (sequence A161953 in the OEIS)

The sequence of base 3 narcissistic numbers starts: 0, 1, 2, 12, 22, 122

The sequence of base 4 narcissistic numbers starts: 0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303 (sequence A010344 and A010343 in OEIS)

In base 2, the only narcissistic numbers are 0 and 1.

The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the kth powers of a k digit number in base b is

and if k is large enough then

in which case no base b narcissistic number can have k or more digits. Setting b equal to 10 shows that the largest narcissistic number in base 10 must be less than 1060.[1]

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.[1]

Clearly, in all bases, all one-digit numbers are narcissistic numbers.

A base b has at least one two-digit narcissistic number if and only if b2 + 1 is not prime, and the number of two-digit narcissistic numbers in base b equals , where is the number of positive divisors of n.

Every base b ≥ 3 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)

Unlike narcissistic numbers, no upper bound can be determined for the size of PDIs in a given base, and it is not currently known whether or not the number of PDIs for an arbitrary base is finite or infinite.[2]

Perfect digital invariant[edit]

Let be a natural number. We define the perfect digital invariant function for base and power 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 perfect digital invariant if it is a fixed point for , which occurs if .[2][7] and are trivial perfect digital invariants for all and , all other perfect digital invariants are nontrivial perfect digital invariants.

For example, the number 4150 in base is a perfect digital invariant with , because .

A natural number is a sociable digital invariant if it is a periodic point for , where for a positive integer , and forms a cycle of period . A perfect digital invariant is a sociable digital invariant with , and a amicable digital invariant is a sociable digital invariant with .

All natural numbers are preperiodic points for , regardless of the base. This is because if , , so any will satisfy until . There are a finite number of natural numbers less than , so the number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point.

Numbers in base lead to fixed or periodic points of numbers .

is the digit sum. The only perfect digital invariants are the single-digit numbers in base , and there are no periodic points with prime period greater than 1.

reduces to , as for any power , and .

Perfect digital invariants of [edit]

By definition, any two-digit perfect digital invariant for with natural number digits has to satisfy the quadratic Diophantine equation .

There are no three-digit perfect digital invariants for .

b = m2k + m + k[edit]

Let be positive integers and the number base . Then:

  • is a perfect digital invariant for for all
  • is a perfect digital invariant for for all .

The table below gives the number base (in decimal) and the perfect digital invariants (in the number base) for every .

Bases and Perfect digital invariants
1 2 3 4
1 3, 12, 22 5, 23, 33 7, 34, 44 9, 45, 55
2 7, 13, 63 12, 25, A5 17, 37, E7 22, 49, I9
3 13, 14, C4 23, 27, L7 33, 3A, UA 43, 4D, [39]D
4 21, 15, K5 38, 29, [36]9 55, 3D, [52]D 72, 4H, [68]H

b = Tk + 2 + (k2 + 4)m[edit]

Let be positive integers and the number base , where the triangular number . Then:

  • is a perfect digital invariant for for all .
  • is a perfect digital invariant for for all .

The table below gives the number base (in decimal) and the perfect digital invariants (in the number base) for every .

Bases and Perfect digital invariants
1 2 3 4
1 3, 22, 12 5, 23, 33 8, 24, 64 12, 25, A5
2 8, 64, 24 13, 67, 77 21, 6A, FA 32, 6D, QD
3 13, A6, 36 21, AB, BB 34, AG, OG 52, AL, [42]L
4 18, E8, 48 29, EF, FF 57, EM, XM 72, ET, [58]T

Perfect digital invariants of [edit]

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.

— G. H. Hardy, A Mathematician's Apology

By definition, any four-digit perfect digital invariant for with natural number digits , , , has to satisfy the quartic Diophantine equation . However, has to be equal to 0, 1, 2 for any , because the maximum value can take is . As a result, there are actually three related cubic Diophantine equations to solve

when
when
when

We take the first case, where .

b = 3k + 1[edit]

Let be a positive integer and the number base . Then:

  • is a perfect digital invariant for for all .
  • is a perfect digital invariant for for all .
  • is a perfect digital invariant for for all .
Perfect digital invariants
1 4 130 131 203
2 7 250 251 305
3 10 370 371 407
4 13 490 491 509
5 16 5B0 5B1 60B
6 19 6D0 6D1 70D
7 22 7F0 7F1 80F
8 25 8H0 8H1 90H
9 28 9J0 9J1 A0J

b = 6k + 4[edit]

Let be a positive integer and the number base . Then:

  • is a perfect digital invariant for for all .
Perfect digital invariants
0 4 021
1 10 153
2 16 285
3 22 3B7
4 28 4E9

Perfect digital invariants and cycles of for specific and [edit]

All numbers are represented in base .

Nontrivial perfect digital invariants Cycles Comments
2 3 12, 22 2 → 11 → 2
  • All terminate at the fixed point 22 or at the cycle starting with .
2 4
2 5 23, 33 4 → 31 → 20 → 4
2 6 5 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5
2 7 13, 34, 44, 63 2 → 4 → 22 → 11 → 2

16 → 52 → 41 → 23 → 16

2 8 24, 64

4 → 20 → 4

5 → 31 → 12 → 5

15 → 32 → 15

2 9 45, 55 58 → 108 → 72 → 58

75 → 82 → 75

2 10 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4
2 11 56, 66 5 → 23 → 12 → 5

68 → 91 → 75 → 68

2 12 25, A5

5 → 21 → 5

8 → 54 → 35 → 2A → 88 → A8 → 118 → 56 → 51 → 22 → 8

18 → 55 → 42 → 18

68 → 84 → 68

2 13 14, 36, 67, 77, A6, C4 28 → 53 → 28

79 → A0 → 79

98 → B2 → 98

2 14 1B → 8A → BA → 11B → 8B → D3 → CA → 136 → 34 → 1B

29 → 61 → 29

2 15 78, 88 2 → 4 → 11 → 2

8 → 44 → 22 → 8

15 → 1B → 82 → 48 → 55 → 35 → 24 → 15

2B → 85 → 5E → EB → 162 → 2B

4E → E2 → D5 → CE → 17A → A0 → 6A → 91 → 57 → 4E

9A → C1 → 9A

D6 → DA → 12E → D6

2 16 D → A9 → B5 → 92 → 55 → 32 → D
3 3
  • All terminate at the cycle starting with .
3 4
  • All terminate at .
  • All terminate at or .
3 6

3 10

  • (sequence A046197 in the OEIS)
  • All terminate at .
  • All terminate at or .
4 3

121 → 200 → 121

122 → 1020 → 122

  • All terminate at the cycle starting with .
4 4 3 → 1101 → 3
4 6

214 → 1133 → 432 → 1345 → 4243 → 2345 → 4310 → 1322 → 310 → 214

3 → 213 → 242 → 1200 → 25 → 2545 → 11014 → 1111 → 4 → 1104 → 1110 → 3

5350 → 10055 → 5443 → 5350

  • Only done for
4 10

2178 → 6514 → 2178

1138 → 4179 → 9219 → 13139 → 6725 → 4338 → 4514 → 1138

  • Only done for
5 3
  • All terminate at .
5 4

3 → 3303 → 23121 → 10311 → 3312 → 20013 → 10110 → 3

3311 → 13220 → 10310 → 3311

  • All terminate at the cycle starting with .
  • All terminate at .
  • All terminate at or the cycle starting with .

Extension to negative integers[edit]

Perfect digital invariants can be extended to the negative integers by use of a balanced base to represent each integer.

Balanced ternary[edit]

In balanced ternary, the digits are 1, −1 and 0. This results in the following:

  • With odd powers , reduces down to digit sum iteration, as , and .
  • With even powers , indicates whether the number is even or odd, as the sum of each digit will indicate divisibility by 2 if and only if the sum of digits ends in 0. As and , for every pair of digits 1 or −1, their sum is 0 and the sum of their squares is 2.

Related concepts[edit]

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 :

,

where is the number of digits in the number in base , and is the value of each digit of the number. is a fixed point of . With this wider definition narcisstic numbers include:

Sums of unary operators on digits[edit]

where is a unary operator or a function, is the number of digits in the number in base , and is the value of each digit of the number.

Products of unary operators on digits[edit]

where is a unary operator or a function, is the number of digits in the number in base , and is the value of each digit of the number.

  • Numbers (sequence A282782 in the OEIS) such that:

Functions of digit sums and products[edit]

where is a unary operator or a function, is the number of digits in the number in base , and is the value of each digit of the number.

  • Dudeney numbers (sequence A061209 in the OEIS) :
  • Numbers (sequence A282693 in the OEIS) such that:
  • Sum-product numbers (sequence A038369 in the OEIS) :

Other[edit]

References[edit]

  1. ^ a b c Weisstein, Eric W. "Narcissistic Number". MathWorld.
  2. ^ a b c 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)
  7. ^ PDIs by Harvey Heinz
  8. ^ Rose, Colin (2005), Radical Narcissistic Numbers, Journal of Recreational Mathematics, 33(4), pages 250-254.
  • 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[edit]