27;} 



On the Representations of a Number as the Sum of Consecutive 



Integers. 



By T. E. Mason. 



(Abstract.) 

 Theorem : 



If we define a series of consecutive integers so as to include zero and 

 negative numbers and if we consider a number itself as a series of con- 

 secutive integers with one term, then a number 



m = 2« . p/'' . Pa"' p,.">-, 



where the p's are the odd prime factors of m and the «'s the power to 

 which they occur, may be expressed as the sum of a series of consecutive 

 integers in 



2(«, + l) («2 + l) («r+l) 



ways. When m = 2" it may be so expressed in two ways. 



One-half of the total number of series will have an even number of 

 terms and one-half will have an odd number of terms. 



One-half of the total number of series will consist of all positive terms 

 and one-half the number of series will contain zero or zero and negative 

 terms. 



We shall now apply this theorem to express 15 as the sum of con- 

 secutive integers. 



15=3x5. 

 The number of series will be 



2(1+1) (1+1)=S. 



Series. 



15 

 4 + 5 + 6 

 1+2+3+4+5 

 -6-5-4-3-2-1+0+1+2+3+4+5+6+7+8 

 7 + 8 

 0+1+2+3+4+5 

 _3_2- 1+0+1 +2+3+4+5+6 

 -14-13.... -4- 3 -2 -1+0 + 1+2 + 3 + 4 + 5+.... +14 + 15 

 //( (liana Univcrsitij, 

 Novemher, 1911. 

 [18—29034] 



