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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.
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 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:
Sociable Dudeney numbers and amicable Dudeney numbers are the powers of their respective roots. The number of iterations i {\displaystyle i} needed for F p , b i ( n ) {\displaystyle F_{p,b}^{i}(n)} to reach a fixed point is the Dudeney function's persistence of n {\displaystyle n} , and undefined if it never reaches a fixed point.
A semiperfect number that is not divisible by any smaller semiperfect number is called primitive. Every number of the form 2 m p for a natural number m and an odd prime number p such that p < 2 m+1 is also semiperfect. In particular, every number of the form 2 m (2 m+1 − 1) is semiperfect, and indeed perfect if 2 m+1 − 1 is a Mersenne prime.
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers, but it is unknown whether there exist odd perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by ...
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.