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Full text of "History Of The Theory Of Numbers - I"

296                        HlSTOKY OF THE THEOKY OF  NUMBERS.                   [CHAP, X
where A and B denote the number of positive and negative terms respectively, not counting o-(O) =n as a term;
+3{(n-6)<r(n-6)+(n-8)(r(n-8) + (n-<r(n-2)+cr(n-4)}
+ (!2+32+52)
<r(n-6)+(r(n-8)+o-(n-10)} + ...   .     (n odd).
He reproduced his64 formulas for 0(n) and E(ri) . He announced (ibid., p. 86) the completion of tables of the values of #(n), r(n), <r(n) up to n = 3000, and inverse tables.
Mobius68 obtained certain results on the reversion of series which were combined by J. W. L. Glaisher69 into the general theorem: Let a, b, ... be distinct primes; in terms of the undefined quantities ea, ebj . . ., let en = eaaeb* ... if n = a'fo*. . . , and let el = 1 . Then, if
where n ranges over all products of powers of a, 6, . . ., we have
where v ranges over the numbers without square factors and divisible by no prime other than a, &,..., while r is the number of the prime factors of v. Taking
Glaisher obtained the formula of H. J. S. Smith38 and
Using the same/, but taking e2=0, ep=pr, when p is an odd prime, he proved that, if Ar(n) is the sum of the rth powers of the odd divisors of n,
Ar(rc) -ZA,^) +24^) - ... =0 or n',
according as n is even or odd.   In the latter case, it reduces to Smith's.
If A'r(n) is the sum of the rth powers of those divisors of n whose complementary divisors are odd, while Er(ri) [or E'r(ri)] is the excess of the sum of the rth powers of those divisors of n which [whose complementary divisors] are of the form 4m -f 1 over the sum of the rth powers of those divisors which [whose complementary divisors] are of the form 4m +3,
6*Jour. fur Math., 9, 1832, 105-123; Werke, 4, 591. "London, Ed. Dublin Phil. Mag., (5), 18, 1884, 518-540.