A091818 -id:A091818 - cp's OEIS frontend

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

Patrick De Geest, May 15 1998

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

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

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

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

Column 2 of A122934.
Cf. A003238, A001065, A027749, A027751 (list of proper divisors).
Cf. A001620, A039653, A051778, A070824, A091818.

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

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

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

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

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

{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

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

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

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

A294015 Sum of the even divisors of 2n, minus the (n-1)st odd number.

Original entry on oeis.org

1, 3, 3, 7, 3, 13, 3, 15, 9, 17, 3, 33, 3, 21, 19, 31, 3, 43, 3, 45, 23, 29, 3, 73, 13, 33, 27, 57, 3, 85, 3, 63, 31, 41, 27, 111, 3, 45, 35, 101, 3, 109, 3, 81, 67, 53, 3, 153, 17, 87, 43, 93, 3, 133, 35, 129, 47, 65, 3, 217, 3, 69, 83, 127, 39, 157, 3, 117, 55, 149, 3, 247, 3, 81, 99, 129, 39, 181, 3, 213, 81, 89, 3, 281

Offset: 1

Omar E. Pol, Oct 28 2017

nonn

a(n) = 3 if and only if n is prime.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Partial sums give A294016.
Cf. A000203, A013661, A005408, A074400, A235796, A294017.
Cf. A001065, A091818.

a[n_] := 2*(DivisorSigma[1, n] - n) + 1; Array[a, 100] (* Amiram Eldar, Mar 30 2024 *)

a(n) = 2*sigma(n) - 2*n + 1; \\ Michel Marcus, Oct 29 2017

a(n) = A074400(n) - A005408(n-1) = 2*A000203(n) - 2*n + 1 = A000203(n) - A235796(n).
Sum_{k=1..n} a(k) = (Pi^2/6 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 30 2024
a(n) = 2*A001065(n) + 1 = A091818(n) + 1. - Omar E. Pol, Dec 01 2024

A279812 Let s(k) denote the sum of the even proper divisors of k. The sequence lists the pairs of numbers (x, y) such that s(x) = y and s(y) = x.

Original entry on oeis.org

440, 568, 2368, 2420, 5240, 5848, 10040, 11128, 12464, 12736, 21488, 21712, 24570, 29190, 34592, 36832, 126040, 133856, 133984, 134190, 139230, 142290, 152168, 159500, 175266, 177460, 200970, 244530, 244736, 246304, 248310, 279630, 283328, 284620, 306352, 337460

Offset: 1

Michel Lagneau, Dec 19 2016

nonn

"Pseudo-amicable" numbers: pairs of numbers (x, y) such that A091818(x) = y and A091818(y) = x.
A pair of numbers x and y is called "pseudo-amicable" if the sum of the proper even divisors of either one is equal to the other. The smallest pair is x = 440, y = 568.
The sequence lists the numbers in increasing order. Note that the pairs (x, y) are not adjacent to each other in the list. For instance, the pair (126040, 152168) are "pseudo-amicable" numbers.
For ordered pairs see A279950.

			The pair (a(1), a(2)) = (440, 568) is in the sequence because the even proper divisors of 440 are 2, 4, 8, 10, 20, 22, 40, 44, 88, 110, 220 with sum = 568 and the proper divisors of 568 are 2, 4, 8, 142, 284 with sum = 440.

Cf. A063990, A074400, A091818, A279950.

s[n_]:=2*(DivisorSigma[1,n/2]-n/2);P[n_]:=If[Nest[s,n,2]==n&&!s[n]==n,True,False];Select[Range[10^6],P[#]&]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A294015 Sum of the even divisors of 2n, minus the (n-1)st odd number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A279812 Let s(k) denote the sum of the even proper divisors of k. The sequence lists the pairs of numbers (x, y) such that s(x) = y and s(y) = x.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica