ON MATHEMATICAL TABLES. 307 



Thus in a table of the form rr + i^, a number of this form has a factor a + hi 

 (i= V —1 as usual) ; and the table in fact shows the complex factor a + bi 

 of the number in question : a well arranged table would give all the prime 

 complex factors a + bi of the number. But as to this more hereafter ; at pre- 

 sent we are concerned with the real theory only, not with any complex theory. 



3. Connected with a factor-table we have (1) Table of the number of less 

 relative primes ; viz. such a table would show for every number the number 

 of inferior integers having no common factor with the number itself. The 

 formula is a weU-known one : for a number 'N=a°-b^cy . . . , (a, b, . . . the di- 

 stinct prime factors of N), the number of less relative primes is 



B7(N), =««-16^-' . . . (rt-l)(6_l) , . . , 



or, what is the same thing, =Nfl )\^~ Tj ' " ^ small table (N=l to 



100), occupying half a page, is given, 



Euler, Op. Arith. Col. t. ii. p. 128; viz. this is 7rl = 0, x2=l, . . . ttIOO 

 =40. 



4. But it would be interesting to have such a table of the same extent with 

 the proposed factor-table. The table might be of like form ; for instance, 



No . of less relative Primes Table 1 to 500 



01234567 89 

 29 ~TT2 I 192 I 144 j 292 | 84 | 232 | 144 | 198 | 148 | 264 | 



and it would be of still greater interest to have an inverse table showing the 

 values of N which belong to a given value of ar(N) ; for instance, 



where, observe, that ot is of necessity even. 



5. Again, connected with a factor-table, we have (2) Table of the Sum of 

 the divisors of a Number. The formula is also a weU-known one ; for a 

 number N=fr 5^. . . , (a, 6 the distinct prime factors of N), the required sum 

 CN is =(l + « . ., + a'^)(l + b ... +b^), ..., or, what is the same thing, 



aa+i— 1 6^+1 — 1 



= = — • -T — ^ — . . . , where, observe, that the number itself is reckoned 



rt— 1 6—1 ' ' 



as a divisor. 



6. Such a table was required by Euler in his researches on Amicable 

 Numbers (see 2^0^^) No. 1(J), and he accordingly gives one of a considerable 

 extent, 



Euler, Op. Arith. CoU. t. i. pp. 104-109. 



It is to be remarked that inasmuch as IN is obviously = jrt" j b^ . . . , the 



function need only be tabulated for the different integer powers a* of each 

 prime number «. The range of Euler's table is as foUows : — 



x2 



