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CHAP.X]                       SUM AND NlJMBEK OF DlVISORS.                             313
former case v=7 (mod 8).   Also the left member of the third formula equals
when v is odd, provided E(Q) = 1/4.    If A'(n) denotes the sum of those divisors of n whose complementary divisors are odd,
A'(n)-2A'(n~l)+2A\'n-4)-2A'(n-9)-|-. . . =0 or (-I)*"1*,
according as n is not or is a square.    [CL Lipschitz.84]   Since A'(n) = <j(n) for ?i odd, we deduce a formula involving <r's and A"s. M. Lerch117 proved (11) and
if /P(n) =S/(d), d ranging over the divisors of n.
K. Th. Vahlen118 proved Liouville's23 formula and Jacobi's10 result.
A. P. Minin119 proved that 2, 8, 9, 12, 18, Sq and 12p (where q is a prime >2, p a prime >3) are the only numbers such that each is divisible by the number of its divisors and the quotient is a prime. Minin120 found that 1, 3, 8, 10, 18, 24 and 30 are the only numbers N for which the number of divisors equals the number of integers <N and prime to N.
M. Lerch121 considered the number x(a, &) and sum X(a, 6) of the divisors ^b of a, proved his100 final formula (17) and
c                                          c
2X(ra an, a) = Sa(x(m an, ri)\l/(m  an, a)},
al                          a"!
(18)             S Mm -an, ~ ) = 2 xfa-an, rn),      c^j^1- 1-
0=0   \             ' /     ao                                 L   n   j
If 5 ranges over the divisors of n,
"am, n)} =S(, m; a),
He quoted (p. 8) from a letter to him from Chr. Zeller that 2 a\f/(m a, a)
equals the sum of the remainders obtained on dividing m by the integers . M. Lerch122 proved that
 crn, n)  S -- p |>
\    x  /                  \    irm"~n rm-f-n~l
-p-pn, cr) =Sx(w-p~o-n, n)~"                      '
117Casopis, Prag, 21, 1892, 90-95, 185-190 (in Bohemian).   Cf. Jahrbuch Fortschritte Math.,
24, 1892; 186-7.
U8Jour. ffir Math., 112, 1893, 29. "'Math. Soc. Moscow, 17, 1893, 240-253. 120/6td., 17, 1894, 537-544. mPrag Sitzungsberichte (Math.), 1894, No. 11. 2/6td., No. 32.