Search results
Results from the WOW.Com Content Network
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:
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 ...
Similarly, the number () is odd if and only if n is a square or twice a square. [9] We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, that is, the divisors of n excluding n itself. This function is used to recognize perfect numbers, which are the n such that s(n) = n.
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.
In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number [1]) is a natural number in a given number base that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because = + + +.
A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758: Extravagant numbers: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ... A number that has fewer digits than the number of digits in its prime factorization (including ...
For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon—say an odd perfect number—it is quite easy to write a partially correct program (see box). But to say this program is totally correct would be to assert something currently not known in number theory.