Factorion

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

In number theory, a factorion in a given number base b is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover. ==Definition== Let n be a natural number. For a base b > 1 , we define the sum of the factorials of the digits of n , \operatorname{SFD}_b : \mathbb{N} \rightarrow \mathbb{N} , to be the following: \operatorname{SFD}_b(n) = \sum_{i=0}^{k - 1} d_i!. where k = \lfloor \log_b n \rfloor + 1 is the number of digits in the number in base b , n! is the factorial of n and d_i = \frac{n \bmod{b^{i+1}} - n \bmod{b^{i}}}{b^{i}} is the value of the i th digit of the number. A natural number n is a b -factorion if it is a fixed point for \operatorname{SFD}_b , i.e. if \operatorname{SFD}_b(n) = n . 1 and 2 are fixed points for all bases b , and thus are trivial factorions for all b , and all other factorions are nontrivial factorions. For example, the number 145 in base b = 10 is a factorion because 145 = 1! + 4! + 5! . For b = 2 , the sum of the factorials of the digits is simply the number of digits k in the base 2 representation since 0! = 1! = 1 . A natural number n is a sociable factorion if it is a periodic point for \operatorname{SFD}_b , where \operatorname{SFD}_b^c(n) = n for a positive integer c , and forms a cycle of period c . A factorion is a sociable factorion with c = 1 , and a amicable factorion is a sociable factorion with c = 2 . All natural numbers n are preperiodic points for \operatorname{SFD}_b , regardless of the base. This is because all natural numbers of base b with k digits satisfy b^{k-1} \leq n . Given that each of the k digits is at most b-1 , \operatorname{SFD}_b \leq (b-1)!k . However, when k \geq b , then b^{k-1} > (b-1)!(k) for b > 2 , so any n will satisfy n > \operatorname{SFD}_b(n) until n . There are finitely many natural numbers less than b^b , so the number is guaranteed to reach a periodic point or a fixed point less than b^b , making it a preperiodic point. For b = 2 , the number of digits k \leq n 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 b . The number of iterations i needed for \operatorname{SFD}_b^i(n) to reach a fixed point is the \operatorname{SFD}_b function's persistence of n , and undefined if it never reaches a fixed point. ===b = (m − 1)!=== Let m be a positive integer and the number base b = (m - 1)! . Then: n_1 = mb + 1 is a factorion for \operatorname{SFD}_b for all m\geq 4 . n_2 = mb + 2 is a factorion for \operatorname{SFD}_b for all m\geq 4 . ===b = m! − m + 1=== Let k be a positive integer and the number base b = m! - m + 1 . Then: n_1 = b + m is a factorion for \operatorname{SFD}_b for all m\geq 3 . ==See also== Arithmetic dynamics Dudeney number Happy number Kaprekar's constant Kaprekar number Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number ==References== ==External links== Factorion at Wolfram MathWorld

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