Mr. C. J. Hargreave on the Law of Prime Numbers. 119 



The proposition is, however, useless when the number of 

 primes to be eliminated is considerable ; for the number being n, 

 the number of terms to be found by actual division would be 

 2"+ ^, if it were not that some of them vanish. It may be observed, 

 however, that when the number x is large, the result will not 

 differ materially from 



'0-,^)('-i)-(-i). 



which can be formed at once. 



If the number be of the form A ( jOj xp^ . . . xjo«), the result 

 is simply A.[p-^ — \){p^—\) . . . (jo^ — l). Thus the number of 

 numbers up to 2 . 3 . 5 . 7 . 11 . 13 . 17 . 19, or 9,699,690, not 

 divisible by either 2, 3, 5, 7, 11, 13, 17 or 19, is simply 



2 X 4 X 6 X 10 X 12 X 16 X 18, or 1,658,880. 



It should be observed, that the result is exclusive of the primes 

 Pu P2, &c. themselves, but inclusive of 1. It is scarcely neces- 

 sary to say that the process is of the easiest possible character^ 

 though tedious from the number of divisions to be effected. I 

 have ascertained by it that the number of ordinals up to ten 

 millions not divisible by either 2, 3, 5, 7, 11, 13, 17, 19 or 23, 

 is 1,635,877 j and the corresponding result up to five millions 

 is 817,944. If we suppose this process performed for all the 

 primes up to the square root of Xy the result would lead us at 

 once to the accurate value of Vx ; but, for the reason above stated, 

 the process is not practically applicable except for the purpose of 



of the integer quotients obtained by dividing a given number A^ by every 

 possible combination of a set of primes, m in number, taken n and n 

 together -, and let the series 



be called ^t. 



Then </>(!) will express the number of ordinals not exceeding A^ which 

 do not contain any one of the m primes. Similarly, — ^'(1) will express 

 the number of ordinals which contain one, and no more, of the set of m 



primes ; — <^"(1) the number of ordinals which contain two, and no more, 



1 

 of these primes ; and generally + -i — 00 0*^^''(1) will express the num- 

 ber of ordinals, each of which contains p distinct individuals of the set of 

 primes, and no more. 



Thus up to 10,000, the number of terms not containing 2, 3, 5, 7 or 11, 

 is 2077 ; the number containing only one of this set is 4193 ; the number 

 containing two* distinct members of this set, and no more, is 2819; the 

 number containing exactly three is 809 ; the number containing exactly 

 four is 98 ; and the number of terms containing all the five is 4. 



