Vadim N. Romanov. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 6, (Part -4) June 2017, pp.01-08 www.ijera.com DOI: 10.9790/9622-0706040108 1 | Page Representation of Integer Positive Number as A Sum of Natural Summands Vadim N. Romanov St.-Petersburg, Russia ABSTRACT In this paper the problem of representation of integer positive number as a sum of natural terms is considered. The new approach to calculation of number of representations is offered. Results of calculations for numbers from 1 to 500 are given. Dependence of partial contributions to total sum of number of representations is investigated. Application of results is discussed. Keywords and phrases: Theory of numbers, representation of integer positive number as a sum of natural terms, number of representations, distribution of partial contributions. I. INTRODUCTION. STATEMENT OF THE PROBLEM AND GENERAL RELATIONS The problem of representation of integer positive number as a sum of the natural summands (terms) has a long story. M. Hall in his book [1] provides an overview of works on this subject and gives representations calculated for the numbers from 1 to 100. However the problem is not solved yet. The purpose of the paper is to study some new aspects of this problem, and to obtain an efficient algorithm for the calculation of the number of representations. The problem is formulated in the following form. A positive integer n is given. We want to find the number of its different representations as a sum of the natural terms. Designate ) (n C – number of representations of n as a sum of terms and let n n  – identity representation. Let’s call the number of representations of n as a sum of terms including the identical representation the proper representation ) ( n C S , and without identical representation – the improper representation ) (n C NS . It is obvious 1 ) ( ) (   n C n C S NS . Hereinafter we mean the proper representation and omit the subscript that does not bring misunderstandings. We give the algorithm of calculation of ) (n C for an arbitrary n , and deduce a recurrent relation that allows to reduce the dimensionality of the problem. For arbitrary natural n we have ) ( ) ( ... ) ( ) ( ) ( ) ( 1 3 2 1 n n n n n n C n n             (1) where 1 ) ( 1  n  – number of representations of n as a sum of units: 1 ... 1 1     n ; ) ( 2 n  – number of representations of n as a sum of units and at least one number two; ) ( 3 n  – number of representations of n as a sum of units, deuces and at least one number three, etc. ) ( 1 n n  – number of representations of n as a sum of units, deuces etc. and at least one number ( 1  n ); 1 ) (  n n  – number of identical representations. Hereinafter we call value () k n  the partial contributions. To reduce the dimension of the problem we use the following obvious ratios 1 1 () ( 1) 1 n n      , 2 1 2 () ( 2) ( 2) n n n        , 3 1 2 3 () ( 3) ( 3) ( 3) n n n n           etc. In the general case we have for arbitrary l 1 2 () ( ) ( ) ... ( ) l l n n l n l n l            , (2) where () l n  – number of representations of n as a sum at least one l and numbers smaller l , 1 ( ) n l   – number of representations of n l  as a sum of units: 1 ( ) 1 n l    . 2 ( ) n l   – number of representations of n l  as a sum of units and at least one number two; etc. ( ) l n l   – number of representations of n l  as a sum at least one number l and numbers smaller l . The following relation is deduced from (2) 1 2 () () () ... () nl nl n l l l           . (3) It is clear that for calculations it is better to use (2) at n l l   and (3) at n l l   . Equations (2), (3) provide an opportunity to reduce dimensionality and RESEARCH ARTICLE OPEN ACCESS