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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 ...
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}} .
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.
In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in base 10: 9981 – 1899 ...
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.
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:
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.
Hence the rate of Hamming codes is R = k / n = 1 − r / (2 r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2 r − 1.