Search results
Results from the WOW.Com Content Network
The following program in Python determines whether an integer number is a Munchausen Number / Perfect Digit to Digit Invariant or not, following the convention =. num = int ( input ( "Enter number:" )) temp = num s = 0.0 while num > 0 : digit = num % 10 num //= 10 s += pow ( digit , digit ) if s == temp : print ( "Munchausen Number" ) else ...
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
The examples below implement the perfect digital invariant function for = and a default base = described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number. A simple test in Python to check if a number is happy:
Check if n is a perfect power: if n = a b for integers a > 1 and b > 1, then output composite. Find the smallest r such that ord r (n) > (log 2 n) 2. If r and n are not coprime, then output composite. For all 2 ≤ a ≤ min (r, n−1), check that a does not divide n: If a|n for some 2 ≤ a ≤ min (r, n−1), then output composite.
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] Euclid proved c. 300 BCE that every Mersenne prime M p = 2 p − 1 has a corresponding perfect number M p × (M p +1)/2 = 2 p − 1 × (2 p − 1). For example, the Mersenne prime 2 2 − 1 = 3 leads to the corresponding perfect number 2 2 − ...
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.
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 b = 10 {\displaystyle b=10} is a perfect digital invariant with p = 5 {\displaystyle p=5} , because 4150 = 4 5 + 1 5 + 5 5 + 0 5 {\displaystyle 4150=4^{5}+1^{5}+5^{5}+0^{5}} .
The number of iterations needed for , to reach a fixed point is the Dudeney function's persistence of , and undefined if it never reaches a fixed point. It can be shown that given a number base b {\displaystyle b} and power p {\displaystyle p} , the maximum Dudeney root has to satisfy this bound: