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%I
%S 0,0,0,1,0,2,0,2,1,2,0,4,0,2,2,3,0,4,0,4,2,2,0,6,1,2,2,4,0,6,0,4,2,2,
%T 2,7,0,2,2,6,0,6,0,4,4,2,0,8,1,4,2,4,0,6,2,6,2,2,0,10,0,2,4,5,2,6,0,4,
%U 2,6,0,10,0,2,4,4,2,6,0,8,3
%N Number of divisors of n which are > 1 and < n (nontrivial divisors).
%C These are sometimes called the proper divisors (see A032741 for the usual meaning of that term)
%C a(n) = number of ordered factorizations of n into two factors, n = 2,3,... If n has the prime factorization n=Product p^e(j), j=1..r, the number of compositions of the vector (e(1), ..., e(r)) equals the number of ordered factorizations of n. Andrews (1998, page 59) gives a formula for the number of m-compositions of (e(1), ..., e(r)) which equals the number f(n,m) of ordered m-factorizations of n. But with m=2 the formula reduces to f(n,2)=d(n)-2=a(n). - A. O. Munagi (amunagi(AT)yahoo.com), Mar 31 2005
%C a(n) = 0 if and only if n is 1 or prime. - _Jon Perry_, Nov 08 2008
%D Andrews, G. E., The Theory of Partitions, Addison-Wesley, Reading 1976; reprinted, Cambridge University Press, Cambridge, 1984, 1998.
%H T. D. Noe, <a href="/A070824/b070824.txt">Table of n, a(n) for n = 1..10000</a>
%H Arnold Knopfmacher and Michael Mays, <a href="http://www.mathematica-journal.com/issue/v10i1/contents/Factorizations/Factorizations_3.html">Ordered and Unordered Factorizations of Integers</a>, The Mathematica Journal, Vol 10 (1).
%F a(n) = A000005(n)-2, n>=2 (with the divisor function d(n)=A000005(n)).
%F a(n) = d(n)-2, where d(n) is the divisor function. E.g. a(12)=4 because 12 has 4 ordered factorizations into two factors: 2*6, 6*2, 3*4, 4*3. - A. O. Munagi (amunagi(AT)yahoo.com), Mar 31 2005
%F G.f.: sum_{k=2..infinity} x^(2k)/(1-x^k). - _Jon Perry_, Nov 08 2008
%e a(12)=4 with the nontrivial divisors 2,3,4,6.
%e a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}. - Peter Luschny, Nov 14 2011
%p seq(numtheory[tau](n)-2,n=2..100); # Munagi, without a(0)
%t Join[{0},Rest[DivisorSigma[0,Range[90]]-2]] (* _Harvey P. Dale_, Jun 23 2012 *)
%Y Cf. A000005, A074206, A032741, A200213.
%K nonn,easy
%O 1,6
%A _Wolfdieter Lang_, May 08 2002
%E a(1)=0 added by Peter Luschny, Nov 14 2011
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