288 HISTORY OF THE THEOBY OF NUMBERS. [CHAP, x and proved the second formula in Liouville's first article and the first two summation formulas of Liouville.29 He proved where 77 = ! or 0 according as 2cr— 1 is divisible by 3 or not. The last two were later generalized by Gegenbauer.80 E. Lionnet33 proved the first two formulas of Liouville.29 J. Liouville34 noted that, if q is divisible by the prime a, C. Sardi35 denoted by An the coefficient of xn in Jacobi's10 series for s3, so that An~0 unless n is a triangular number. From that series he got S(-l)^2p+l)cr{w-p(p+l)/2} = (-l)('+1)/X/3 or 0 p according as n is or is not a triangular number, and -0. This recursion formula determines An hi terms of the <r's, or <r(n) hi terms of the A's. In each case the values are expressed by means of determinants of order n. M. A. Andreievsky36 wrote .AT^i for the number of the divisors of the form 4/1=*=! of n = att6^. . ., where a, 6, ... are distinct prunes. We have «'-o where d ranges over all the divisors of n and the symbols are Legendre's. Evidently B /_1v.' 2 (— ^) = a+l jf a=4Z+l, a'-0\ O / = 0or 1 if a=4Z-l, according as a is odd or even. Hence, if any prime factor 41 — I of n occurs to an odd power, we have A^a+i = A^-I. Next, let n = p1>2a3...g12V?2-.., where each pt- is a prime of the form 41 4-1, each & of the form 4Z — 1. Then 33Nouv. Ann. Math., (2), 7, 1868, 6&-72. ^Jour. de math., (2), 14, 1869, 263-4. ^Giornale di Mat., 7, 1869, 112-5. 36Mat. Sbornik (Math. Soc. Moscow), 6, 1872-3, 97-106 (Russian).