## Exercises and problems in Informatics |

## Please read The Conditions of the Problem Solving Competition.

**I. 58.** A sequence *a*_{1}, *a*_{2}, ..., *a*_{n} of positive integers *each* having *at most* *N* digits is called an *aliquot sequence* of *N* digits, if the sum of positive proper divisors (*i.e. *including 1 but excluding the number itself) of *a*_{i} is *a*_{i+1} (*i*=1,2,...,*n*-1) and that of *a*_{n} is *a*_{1}. Members of an aliquot sequence are called *sociable numbers.* (Thus, the aliquot sequences of length 1 are just the perfect numbers, and aliquot sequences of length 2 are the usual amicable pairs.)

Your program (i58.pas, ...) should read the value of *N* (1\(\displaystyle \le\)*N* \(\displaystyle \le\)8), compute every sociable numbers of *N* digits for which the smallest element is in the interval [*A*,*B*], then write the output into the text file ``i58.out''.

The file containing the output corresponding to parameter values *N*=7, *A*=2, *B*=9 999 999 should be submitted.

*Examples.* Sociable numbers of 3 digits with the smallest element in the interval [200,230] form the well-known amicable pair 220-284.

Sociable numbers of 5 digits with the smallest element in the interval [10000,13000] include 10744-10856, 12285-14595, 12496-14288-15472-14536-14264.

(*10 points*)

**I. 59.** A *regular star polygon* with *N* vertices is obtained by connecting every vertex of a regular *N*-gon with both of its *K*^{th} neighbours.

Write a program (i59.pas, ...) which reads the value of *N* (5\(\displaystyle \le\)*N* \(\displaystyle \le\)100), then displays all *distinct* regular star polygons with *N* vertices.

The *example* shows all 4 distinct regular star polygons with 11 vertices.

(*10 points*)

**I. 60.** Similarly to the concept of *highly composite* numbers (see Problem **I. 55.** in the September 2003 issue), we say that a positive integer *n* \(\displaystyle \in\)[*A*,*B*] is *highly composite with respect to the interval* [*A*,*B*], if the number of divisors of *n* is greater than or equal to that of any positive integers in the interval [*A*,*B*] below *n*.

Prepare your sheet (i60.xls) which - if *A* and *B* (1\(\displaystyle \le\)*A*\(\displaystyle \le\)*B*\(\displaystyle \le\)1000) are given - displays all numbers in the interval [*A*,*B*] highlighting with red all highly composite numbers with respect to that interval.

In the *example* italic letters show the red highlighting.

A=10 | B=25 |

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(*10 points*)