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A070824 Number of divisors of n which are > 1 and < n (nontrivial divisors). 31
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, 2, 0, 10, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

These are sometimes called the proper divisors, but 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 number-of-divisors function d(n) = A000005(n)).

a(n) = d(n)-2, for n>=2, where d(n) is the number-of-divisors 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} 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 *)

a[ n_] := SeriesCoefficient[ Sum[x^(2 k)/(1 - x^k), {k, 2, n/2}], {x, 0, n}]; (* Michael Somos, Jun 24 2019 *)

PROG

(Haskell) a070824 n = if n == 1 then 0 else length $ tail $ a027751_row n -- Reinhard Zumkeller, Dec 03 2014

(PARI) {a(n) = if( n<1, 0, my(v = vector(n, i, i>1)); dirmul(v, v)[n])}; /* Michael Somos, Jun 24 2019 */

(PARI) apply( A070824(n)=numdiv(n+(n<2))-2, [1..90]) \\ M. F. Hasler, Oct 11 2019

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: A308062 A147588 A307409 * A174725 A071459 A319164

Adjacent sequences:  A070821 A070822 A070823 * A070825 A070826 A070827

KEYWORD

nonn,easy,changed

AUTHOR

Wolfdieter Lang, May 08 2002

EXTENSIONS

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

Several minor edits by M. F. Hasler, Oct 14 2019

STATUS

approved

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Last modified October 15 17:24 EDT 2019. Contains 328037 sequences. (Running on oeis4.)