A032741 a(0) = 0; for n > 0, a(n) = number of proper divisors of n (divisors of n which are less than n).

0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 11, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 11, 3, 3, 3, 7, 1, 11, 3, 5, 3, 3, 3, 11, 1, 5, 5

Offset: 0

Author

Patrick De Geest, May 15 1998

Keywords

• nonn
• easy

Comments

Number of d < n which divide n.

Call an integer k between 1 and n a "semi-divisor" of n if n leaves a remainder of 1 when divided by k, i.e., n == 1 (mod k). a(n) gives the number of semi-divisors of n+1. - Joseph L. Pe, Sep 11 2002

a(n+1) is also the number of k, 0 <= k <= n-1, such that C(n,k) divides C(n,k+1). - Benoit Cloitre, Oct 17 2002

a(n+1) is also the number of factors of the n-th degree polynomial x^n + x^(n-1) + x^(n-2) + ... + x^2 + x + 1. Example: 1 + x + x^2 + x^3 = (1+x)(1+x^2) implies a(4)=2.

a(n) is also the number of factors of the n-th Fibonacci polynomial. - T. D. Noe, Mar 09 2006

Number of partitions of n into 2 parts with the second dividing the first. - Franklin T. Adams-Watters, Sep 20 2006

Number of partitions of n+1 into exactly one q and at least one q+1. Example: a(12)=5; indeed, we have 13 = 7 + 6 = 5 + 4 + 4 = 4 + 3 + 3 + 3 = 3 + 2 + 2 + 2 + 2 + 2 = 2 + 11*1.

Differences of A002541. - George Beck, Feb 12 2012

For n > 1: number of ones in row n+1 of triangle A051778. - Reinhard Zumkeller, Dec 03 2014

For n > 0, a(n) is the number of strong divisors of n. - Omar E. Pol, May 03 2015

a(n) is also the number of factors of the (n-1)-th degree polynomial ((x+1)^n-1)/x. Example: for n=6, ((x+1)^6-1)/x = x^5 + 6*x^4 + 15*x^3 + 20*x^2 + 15*x + 6 = (2+x)(1+x+x^2)(3+3x+x^2) implies a(6)=3. - Federico Provvedi, Oct 09 2018

Consider the polynomial P(n,z) = Sum_{i=1..q} d(i)*z^(i-1) where d(1), d(2), ..., d(q) are are the q ordered divisors of n. The sequence lists the numbers of zeros of P(n,z) strictly inside the unit circle. - Michel Lagneau, Apr 06 2025

Examples

			a(6) = 3 since the proper divisors of 6 are 1, 2, 3.
		

References

• André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, 1984, page 5.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Proper divisors.

Crossrefs

Column 2 of A122934.

Cf. A003238, A001065, A027749, A027751 (list of proper divisors).

Cf. A001620, A039653, A051778, A070824, A091818.

Programs

GAP

Concatenation([0],List([1..100],n->Tau(n)-1)); # Muniru A Asiru, Oct 09 2018

Haskell

a032741 n = if n == 0 then 0 else a000005 n - 1
-- Reinhard Zumkeller, Jul 31 2014

Maple

A032741 := proc(n)
    if n = 0 then
        0 ;
    else
        numtheory[tau](n)-1 ;
    end if;
end proc: # R. J. Mathar, Feb 03 2013

Mathematica

Prepend[DivisorSigma[0, Range[99]]-1, 0] (* Jayanta Basu, May 25 2013 *)

PARI

a(n) = if(n<1,0,numdiv(n)-1)

PARI

{a(n)=polcoeff(2*sum(m=1,n\2+1,sumdiv(m,d,log(1-x^(m/d) +x*O(x^n) )^(2*d)/(2*d)!)), n)} \\ Paul D. Hanna, Aug 21 2014

Python

from sympy import divisor_count
def A032741(n): return divisor_count(n)-1 if n else 0 # Chai Wah Wu, Mar 14 2023

Formula

a(n) = tau(n)-1 = A000005(n)-1. Cf. A039653.

G.f.: Sum_{n>=1} x^(2*n)/(1-x^n). - Michael Somos, Apr 29 2003

G.f.: Sum_{i>=1} (1-x^i+x^(2*i))/(1-x^i). - Jon Perry, Jul 03 2004

a(n) = Sum_{k=1..floor(n/2)} A051731(n-k,k). - Reinhard Zumkeller, Nov 01 2009

G.f.: 2*Sum_{n>=1} Sum_{d|n} log(1 - x^(n/d))^(2*d) / (2*d)!. - Paul D. Hanna, Aug 21 2014

Dirichlet g.f.: zeta(s)*(zeta(s)-1). - Geoffrey Critzer, Dec 06 2014

a(n) = Sum_{k=1..n-1} binomial((n-1) mod k, k-1). - Wesley Ivan Hurt, Sep 26 2016

a(n) = Sum_{i=1..n-1} floor(n/i)-floor((n-1)/i). - Wesley Ivan Hurt, Nov 15 2017

a(n) = Sum_{i=1..n-1} 1-sign(i mod (n-i)). - Wesley Ivan Hurt, Sep 27 2018

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

Extensions

Typos in definition corrected by Omar E. Pol, Dec 13 2008