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A070824
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Number of divisors of n which are > 1 and < n (nontrivial divisors).
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13
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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, 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, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 3
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OFFSET
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1,6
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COMMENTS
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These are sometimes called the proper divisors (see A032741 for the usual meaning of that term)
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
a(n) = 0 if and only if n is 1 or prime. - Jon Perry, Nov 08 2008
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REFERENCES
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Andrews, G. E., The Theory of Partitions, Addison-Wesley, Reading 1976; reprinted, Cambridge University Press, Cambridge, 1984, 1998.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).
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FORMULA
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a(n) = A000005(n)-2, n>=2 (with the divisor function d(n)=A000005(n)).
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
G.f.: sum_{k=2..infinity} x^(2k)/(1-x^k). - Jon Perry, Nov 08 2008
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EXAMPLE
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a(12)=4 with the nontrivial divisors 2,3,4,6.
a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}. - Peter Luschny, Nov 14 2011
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MAPLE
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seq(numtheory[tau](n)-2, n=2..100); # Munagi, without a(0)
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MATHEMATICA
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Join[{0}, Rest[DivisorSigma[0, Range[90]]-2]] (* Harvey P. Dale, Jun 23 2012 *)
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CROSSREFS
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Cf. A000005, A074206, A032741, A200213.
Sequence in context: A074398 A144765 A147588 * A174725 A071459 A070288
Adjacent sequences: A070821 A070822 A070823 * A070825 A070826 A070827
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, May 08 2002
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EXTENSIONS
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a(1)=0 added by Peter Luschny, Nov 14 2011
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STATUS
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approved
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