308 



viz. for the several prime numbers from 29 to 997 the table gives f rt, (*«-, 

 and Ca^. And it is to be noticed that the values of the sum are exhibited, 

 decomposed into their prime factors : thus a specimen of the table is 



Num. Summa Divisorum. 



7. The form of the above table is adapted to its particular purpose (the 

 theory of amicable numbers) ; but Euler gives also, 



Euler, Op. Arith. Coll. t. i. p. 147 (in the paper " Observatio de Summis 

 Divisorum," pp. 146-1.54, 1752), a short table of about half a page, 1^=1 

 to 100, of the form Jl = l, J2=3, . ..("100=217. The paper contains on 

 the subject of j N interesting analytical researches which connect themselves 

 with the theory of the Partition of Numbers. 



8. It would be interesting to carry the last-mentioned tabic to the same 

 extent as the proposed factor-table ; and to add to it an inverse table, as sug- 

 gested in regard to the number of less relative primes table. 



9. Perfect JVumhers. — A perfect number is a number which is equal to the 

 sum of its divisors (the number itself not being reckoned as a divisor), e. g. 

 6 = 1+2 + 3: 28 =1 + 2-1-4-1-7-1- 14. Such numbers are indicated by a table 



of the sums of divisors J 6 = 12, J28 = 56, these two being, as appears by 

 the table, art. 7, the only perfect numbers less than 100. 



10. Amicable Numhers. — These are pairs of numbers such that each is equal 

 to the sum of the divisors of the other (not reckoning the other number as 

 a divisor) ; or, what is the same thing, such that each has the same sum of 

 divisors (the number being here reckoned as a divisor); say rA=B, r'B=A ; 

 or, what is the same thing, Ja= J'B(= A + B). Thus 220, 284, 



J'220=(l+2 + 4)(l + 5)(l + ll)-220, =284, 



J'284=(l+2 + 4)(l + 71)-284, =220; 



or, what is the same thing, 



j220=(l + 2 + 4)(l + 5)(l + ll)=504=(l + 2 + 4)(l + 71) = j284, 



11. A catalogue of 61 pairs of numbers is given 



Euler, Op. Arith. Coll. t. i, pp. 144-145 (occupies about one page). The 

 paper, "Be Numeris Amicabilibus," pp. 102-145, contains an elaborate in- 



