Factorion

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In number theory, a factorion in a given number base is a natural number that equals the sum of the factorials of its digits.[1][2][3] The name factorion was coined by the author Clifford A. Pickover.[4]

Definition[edit]

Let be a natural number. We define the sum of the factorial of the digits[5][6] of for base to be the following:

.

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

is the value of each digit of the number. A natural number is a -factorion if it is a fixed point for , which occurs if .[7] and are fixed points for all , and thus are trivial factorions for all , and all other factorions are nontrivial factorions.

For example, the number 145 in base is a factorion because .

For , the sum of the factorial of the digits is simply the number of digits in the base 2 representation.

A natural number is a sociable factorion if it is a periodic point for , where for a positive integer , and forms a cycle of period . A factorion is a sociable factorion with , and a amicable factorion is a sociable factorion with .[8][9]

All natural numbers are preperiodic points for , regardless of the base. This is because all natural numbers of base with digits satisfy . However, when , then for , 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. For , the number of digits for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles for any given base .

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

Factorions for [edit]

b = (k - 1)![edit]

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

  • is a factorion for for all .
  • is a factorion for for all .
Factorions
4 6 41 42
5 24 51 52
6 120 61 62
7 720 71 72

b = k! - k + 1[edit]

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

  • is a factorion for for all .
Factorions
3 4 13
4 21 14
5 116 15
6 715 16

Table of factorions and cycles of [edit]

All numbers are represented in base .

Base Nontrivial factorion (, )[10] Cycles
2
3
4 13 3 → 12 → 3
5 144
6 41, 42
7 36 → 2055 → 465 → 2343 → 53 → 240 → 36
8

3 → 6 → 1320 → 12

175 → 12051 → 175

9 62558
10 145, 40585

871 → 45361 → 871[9]

872 → 45362 → 872[8]

Programming example[edit]

The example below implements the sum of the factorial of the digits described in the definition above to search for factorions and cycles in Python.

def factorial(x):
    total = 1
    for i in range(0, x):
        total = total * (i + 1)
    return total

def SFD(x, b):
    total = 0
    while x > 0:
        total = total + factorial(x % b)
        x = x // b
    return total

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

See also[edit]

References[edit]

  1. ^ Sloane, Neil, "A014080", On-Line Encyclopedia of Integer Sequences
  2. ^ Gardner, Martin (1978), "Factorial Oddities", Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-Of-Mind, Vintage Books, pp. 61 and 64
  3. ^ Madachy, Joseph S. (1979), Madachy's Mathematical Recreations, Dover Publications, p. 167
  4. ^ Pickover, Clifford A. (1995), "The Loneliness of the Factorions", Keys to Infinity, John Wiley & Sons, pp. 169-171 and 319-320 – via Google Books
  5. ^ Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", The Mathematical Gazette, The Mathematical Association, 88 (512): 258–261
  6. ^ Sloane, Neil, "A061602", On-Line Encyclopedia of Integer Sequences
  7. ^ Abbott, Steve (2004), "SFD Chains and Factorion Cycles", The Mathematical Gazette, The Mathematical Association, 88 (512): 261–263
  8. ^ a b Sloane, Neil, "A214285", On-Line Encyclopedia of Integer Sequences
  9. ^ a b Sloane, Neil, "A254499", On-Line Encyclopedia of Integer Sequences
  10. ^ Sloane, Neil, "A193163", On-Line Encyclopedia of Integer Sequences

External links[edit]