Exercises and problems in Informatics
I. 58. A sequence a1, a2, ..., an 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 ai is ai+1 (i=1,2,...,n-1) and that of an is a1. 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.
I. 59. A regular star polygon with N vertices is obtained by connecting every vertex of a regular N-gon with both of its Kth 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.
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.
Send your solutions to the following e-mail address:
Deadline: 13 November 2003