

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
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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 mcompositions of (e(1), ..., e(r)) which equals the number f(n,m) of ordered mfactorizations 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 nth 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/d2),d1] is a partition of n of the prescribed type.  Emeric Deutsch, Nov 03 2015


REFERENCES

Andrews, G. E., The Theory of Partitions, AddisonWesley, 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 numberofdivisors function d(n) = A000005(n)).
a(n) = d(n)2, for n>=2, where d(n) is the numberofdivisors 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)/(1x^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



