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A070824 Number of divisors of n which are > 1 and < n (nontrivial divisors). 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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). - Augustine O. Munagi, Mar 31 2005

a(n) = 0 if and only if n is 1 or prime. - Jon Perry, Nov 08 2008

For n > 2: number of zeros in n-th row of triangle A051778. - Reinhard Zumkeller, Dec 03 2014

a(n) = number of partitions of n in which largest and least parts occur exactly once and their difference is 2. Example: a(12) = 4 because we have [7,5], [5,4,3], [4,3,3,2], and [3,2,2,2,2,1]. In general, if d is a nontrivial divisor of n, then [d+1,{d}^(n/d-2),d-1] is a partition of n of the prescribed type. - Emeric Deutsch, Nov 03 2015

REFERENCES

Andrews, G. E., The Theory of Partitions, Addison-Wesley, Reading 1976; reprinted, Cambridge University Press, Cambridge, 1984, 1998.

LINKS

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).

FORMULA

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. - Augustine O. Munagi, Mar 31 2005

G.f.: sum_{k=2..infinity} x^(2k)/(1-x^k). - Jon Perry, Nov 08 2008

Dirichlet generating function: (Zeta(s)-1)^2 [Mats Granvik May 25 2013]

EXAMPLE

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

MAPLE

0, seq(numtheory[tau](n)-2, n=2..100); # Augustine O. Munagi, Mar 31 2005

MATHEMATICA

Join[{0}, Rest[DivisorSigma[0, Range[90]]-2]] (* Harvey P. Dale, Jun 23 2012 *)

PROG

(Haskell)

a070824 n = if n == 1 then 0 else length $ tail $ a027751_row n

-- Reinhard Zumkeller, Dec 03 2014

CROSSREFS

Cf. A000005, A074206, A032741, A200213.

Cf. First column in the matrix power A175992^2

Row sums of A175992 starting from the second column.

Cf. A027751, A051778.

Column k=2 of A251683.

Sequence in context: A074398 A144765 A147588 * A174725 A071459 A070288

Adjacent sequences:  A070821 A070822 A070823 * A070825 A070826 A070827

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 08 2002

EXTENSIONS

a(1)=0 added by Peter Luschny, Nov 14 2011

STATUS

approved

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Last modified May 26 12:59 EDT 2017. Contains 287095 sequences.