259 



2M7, 21". 23. 89 



2M7, 2". 19 . 683 . 2731 . 8191 



2».31, 2>^43.127t 



2".3S 22».3M27.337 



2^8. 7 . 23 . 233 . 1103 . 2089, 2=«. 7\ 43 . 223 . 7019 . 112303 . 898423 . 616318177 



2". 131071, 2". 174763 . 524289 



23«. 53 . 229 . 8191 . 121369, 2«'. 59. 157 . 43331 . 3033169 . 715827883 . 2147483647 



3^ 11M3, 3M1. 13= 



3«. 137 . 547 . 1093, 3'o. 107 . 3851 



3'. 23. 41, 3'". 232. 79. 107.3851 



5=.7=. 19.31, 5'. 7'. 13 



oM3-. 31=. 61. 83. 331, 5M3M7 



5=.72. 19M27, 5^7M9 



5'. 1\ \ZK 17-. 307 . 467 . 2801, 5^ 1\ 13 . 17 . 71 



If N = ri/i r2/= Tn'" , where ri, r-i,. . . ., r,, are different primes, is a 



multiply perfect number of multiplicity m, then from the formula for the 

 sum of all the divisors of N and the fact that this sum is now supposed to 

 be mN, we have 



n Tj— 1 



m = n 



i = 1 > i 



Therefore in order to prove the accuracy of the rules we have only to show 

 in each case that 



Pi - 1 Qi - 1 



n— ^^ = n ^' 



Pi (Pi-1) Qi (Qi-l) 



The verification is not carried out. 



II. // n p,oi (nil) and n q^ ,^i (n\.{) (in either order) are a pair of factor sets 

 and viultiplicitij from the list below and if a multiply perfect number Ni of mul- 

 tiplicity mi contains th3 factor II Pi"i without containing either any factor P,"'+l 

 or any factor q^ different from every pi; then the number 



N,nq,3i 



No 



Pi" 



This pair Is due to i:)escaifos. 



