A070824 Number of divisors of n which are > 1 and < n (nontrivial divisors).

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

Offset: 1

Author

Wolfdieter Lang, May 08 2002

Keywords

• nonn
• easy

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

Absolute values of the inverse Möbius transform of (-1)^prime(n), n >= 2. - Wesley Ivan Hurt, Jun 22 2024

Examples

			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
		

References

• George E. Andrews, 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
• Rahul Chhimpa and Avinash Chand Yadav, Scaling behavior in the number theoretic division model of self-organized criticality, arXiv:2410.05699 [cond-mat.stat-mech], 2024. See p. 2.
• Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1), 2006 (Wayback Machine link); ResearchGate link.

Crossrefs

Cf. A000005, A001620, A074206, A032741, A200213.

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.

Programs

Haskell

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

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 *)

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

Python

from sympy import divisor_count
def A070824(n): return 0 if n == 1 else divisor_count(n)-2 # Chai Wah Wu, Jun 03 2022

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

Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - 3)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

a(n) = abs( Sum_{d|n} (-1)^prime(d) ), n >= 2 with a(1) = 0. - Wesley Ivan Hurt, Jun 22 2024

a(n) = Sum_{k=2..n-1} floor(n/k) - floor((n-1)/k), see Chhimpa and Yadav. - Stefano Spezia, Oct 13 2024

Extensions

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

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