273 



On the Representations op a Number as the Sum op Consecutive 



Integers. 



By T. E. Mason. 



(Abstract.) 

 '■J'heorem : 



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 witb one term, tben a number 



m = 2« . pi'" . p/'> p,.">-, 



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

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

 integers in 



2(", + l) («.> + 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 numlier of series will be 



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



Sei'ies. 



15 

 4 + 5-f6 

 1 + 2-^3-1-4 + 5 

 -6-5-4-3-2- 1+0 + 1 + 2 + 3 + 4 + 5 + 6 -h7 + 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 



