%I A032741
%S A032741 0,0,1,1,2,1,3,1,3,2,3,1,5,1,3,3,4,1,5,1,5,3,3,1,7,2,3,3,5,1,7,1,5,3,3,
%T A032741 3,8,1,3,3,7,1,7,1,5,5,3,1,9,2,5,3,5,1,7,3,7,3,3,1,11,1,3,5,6,3,7,1,
%U A032741 5,3,7,1,11,1,3,5,5,3,7,1,9,4,3,1,11,3,3,3,7,1,11,3,5,3,3,3,11,1,5,5
%N A032741 a(0) = 0; for n > 0, a(n) = number of proper divisors of n.
%C A032741 Number of d < n which divide n.
%C A032741 Call an integer k between 1 and n a "semi-divisor" of n if n leaves a 
               remainder of 1 when divided by k, i.e. n == 1 (mod k). a(n) gives 
               the number of semi-divisors of n+1. - Joseph L. Pe (joseph_l_pe(AT)hotmail.com), 
               Sep 11 2002
%C A032741 a(n+1) is also the number of k, 0<=k<=n-1, such that C(n,k) divides C(n,
               k+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 17 2002
%C A032741 Also the number of factors of the n-th degree polynomial x^n + x^(n-1) 
               + x^(n-2) + ... + x^2 + x + 1.
%C A032741 Also the number of factors of the n-th Fibonacci polynomial. - T. D. 
               Noe (noe(AT)sspectra.com), Mar 09 2006
%C A032741 Number of partitions of n into 2 parts with the second dividing the first. 
               - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 20 2006
%H A032741 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ProperDivisor.html">Proper divisors</a>
%F A032741 G.f.: Sum_{n>0} x^(2n)/(1-x^n). - Michael Somos, Apr 29, 2003
%F A032741 G.f.: sum(i=1, oo, (1-x^i+x^(2*i))/(1-x^i)) - Jon Perry (perry(AT)globalnet.co.uk), 
               Jul 03 2004
%F A032741 a(n) = SUM(A051731(n-k,k): 1 <= k <= n/2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 01 2009]
%e A032741 a(6) = 3 since the proper divisors of 6 are 1, 2, 3.
%o A032741 (PARI) a(n) = if(n<1,0,numdiv(n)-1)
%Y A032741 Equals tau(n) - 1 = A000005(n) - 1. Cf. A039653.
%Y A032741 Column 2 of A122934.
%Y A032741 Cf. A003238, A001065, A027751 (list of proper divisors).
%Y A032741 Sequence in context: A128199 A036459 A079167 this_sequence A046051 A025812 
               A109698
%Y A032741 Adjacent sequences: A032738 A032739 A032740 this_sequence A032742 A032743 
               A032744
%K A032741 nonn,easy,new
%O A032741 0,5
%A A032741 Patrick De Geest (pdg(AT)worldofnumbers.com), May 15 1998.
%E A032741 Typos in definition corrected by Omar E. Pol, Dec 13 2008