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In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k , a number n is called k -perfect (or k -fold perfect) if the sum of all positive divisors of n (the divisor function , σ ( n )) is equal to kn ; a number is thus perfect if and ...
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.
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 arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: n 4 = n × n × n × n. Fourth powers are also formed by multiplying a number by its cube.
The smallest friendly number is 6, forming for example, the friendly pair 6 and 28 with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with abundancy 2 are also known as perfect numbers. There are several unsolved ...
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of ...
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In 1957 he, with Bernhard Hornfeck, gave an asymptotic estimate for the density of perfect numbers. [6] In 1959 Wirsing gave an asymptotic estimate for the density of multiply perfect numbers. [7] He gave in 1962 an elementary proof of a sharpened form of the prime-number theorem (with remainder).