Superperfect number

A natural number n is referred to as super perfect number if the sum of the divisors of the sum of its divisors is twice as large as the original number n used to write as a notation for the divisor sum function, we can define as follows:

In contrast, the well-known perfect numbers satisfy the equation, the question whether a number is super perfect, arises in the study of iterated divisor sum function ( see also content chain, here, however, the picture is iterated ).

Examples and properties

The number 6, the splitter 1, 2, 3 and 6, the sum of these numbers is 12, the splitter 12, in turn, are 1, 2, 3, 4, 6 and 12, the sum is 28. Because 28 ≠ 2.6 is 6 not super perfect number. More computational examples are:

The first super perfect numbers are 2, 4, 16, 64, 4096, 65536, 262144, ... ( sequence A019279 in OEIS ).

Every even perfect number has the super shape with a Mersenne prime number (example: 16 is super perfect and 31 is a Mersenne prime ). Conversely, every Mersenne prime number is an even super perfect number. Whether there are odd super perfect numbers, is not known.

Generalization

Super Perfect numbers are - just like the perfect numbers - Examples of figures of the upper class of (m, k) -super perfect numbers, which are defined as follows:

Perfect numbers are thus (1,2)- super perfect and super perfect numbers ( 2,2)- super perfect. Cohen / te Riele think it possible that any number (m, k) is super perfect for suitable m and k