Factorion

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

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

b = (k - 1)!

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

  • is a factorion for for all .
Proof 

Let the digits of be , and . Then

Thus is a factorion for for all .

  • is a factorion for for all .
Proof 

Let the digits of be , and . Then

Thus is a factorion for for all .

Factorions
464142
5245152
61206162
77207172

b = k! - k + 1

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

  • is a factorion for for all .
Proof 

Let the digits of be , and . Then

Thus is a factorion for for all .

Factorions
3413
42114
511615
671516

Table of factorions and cycles of

All numbers are represented in base .

Base Nontrivial factorion (, )[10] Cycles
2
3
4133 → 12 → 3
5144
641, 42
736 → 2055 → 465 → 2343 → 53 → 240 → 36
8

3 → 6 → 1320 → 12

175 → 12051 → 175

962558
10145, 40585

871 → 45361 → 871[9]

872 → 45362 → 872[8]

Programming example

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: int) -> int:
    total = 1
    for i in range(0, x):
        total = total * (i + 1)
    return total

def sfd(x: int, b: int) -> int:
    """Sum of the factorial of the digits."""
    total = 0
    while x > 0:
        total = total + factorial(x % b)
        x = x // b
    return total

def sfd_cycle(x: int, b: int) -> List[int]:
    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

References

  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, ISBN 9780394726236
  3. Madachy, Joseph S. (1979), Madachy's Mathematical Recreations, Dover Publications, p. 167, ISBN 9780486237626
  4. Pickover, Clifford A. (1995), "The Loneliness of the Factorions", Keys to Infinity, John Wiley & Sons, pp. 169–171 and 319–320, ISBN 9780471193340 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, doi:10.1017/S0025557200174996, JSTOR 3620841
  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, doi:10.1017/S002555720017500X, JSTOR 3620842
  8. Sloane, Neil, "A214285", On-Line Encyclopedia of Integer Sequences
  9. Sloane, Neil, "A254499", On-Line Encyclopedia of Integer Sequences
  10. Sloane, Neil, "A193163", On-Line Encyclopedia of Integer Sequences
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