A244049 Sum of all proper divisors of all positive integers <= n.

0, 0, 0, 2, 2, 7, 7, 13, 16, 23, 23, 38, 38, 47, 55, 69, 69, 89, 89, 110, 120, 133, 133, 168, 173, 188, 200, 227, 227, 268, 268, 298, 312, 331, 343, 397, 397, 418, 434, 483, 483, 536, 536, 575, 607, 632, 632, 707, 714, 756, 776, 821, 821, 886, 902

Offset: 1

Author

Omar E. Pol, Jun 24 2014

Keywords

• nonn
• easy

Comments

The proper divisors of n are all divisors except 1 and n itself. Therefore noncomposite numbers have no proper divisors.

For the sum of all aliquot divisors of all positive integers <= n see A153485.

For the sum all divisors of all positive integers <= n see A024916.

a(n) = a(n - 1) if and only if n is prime.

For n >= 3 a(n) equals the area of an arrowhead-shaped polygon formed by two zig-zag paths and the Dyck path described in the n-th row of A237593 as shown in the Links section. Note that there is a similar diagram of A153485(n) in A153485. - Omar E. Pol, Jun 14 2022

Examples

			a(4) = 2 because the only proper divisor of 4 is 2 and the previous n contributed no proper divisors to the sum.
a(5) = 2 because 5 is prime and contributes no proper divisors to the sum.
a(6) = 7 because the proper divisors of 6 are 2 and 3, which add up to 5, and a(5) + 5 = 2 + 5 = 7.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Omar E. Pol, Illustration of initial terms with arrowhead-shaped polygons.

Crossrefs

Partial sums of A048050.

Cf. A000040, A000203, A000290, A001065, A004125, A008578, A024916, A027750, A034856, A153485, A161680, A237591, A237593, A245092, A262626.

Programs

Mathematica

propDivsRunSum[1] := 0; propDivsRunSum[n_] := propDivsRunSum[n] = propDivsRunSum[n - 1] + (Plus@@Divisors[n]) - (n + 1); Table[propDivsRunSum[n], {n, 60}] (* Alonso del Arte, Jun 30 2014 *)
Accumulate[Join[{0},Table[Total[Most[Divisors[n]]]-1,{n,2,60}]]] (* Harvey P. Dale, Aug 12 2016 *)
Accumulate[Join[{0}, Table[DivisorSigma[1, n] - n - 1, {n, 2, 55}]]] (* Amiram Eldar, Jun 18 2022 *)

PARI

a(n) = sum(k=2, n, sigma(k)-k-1); \\ Michel Marcus, Mar 30 2021

Python

from math import isqrt
def A244049(n): return ((-n*(n+3)-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1)+1 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = A024916(n) - A034856(n).

a(n) = A153485(n) - n + 1.

G.f.: (1/(1 - x))*Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017

a(n) = A161680(n-1) - A004125(n). - Omar E. Pol, Mar 25 2021

a(n) = A000290(n) - A034856(n) - A004125(n). - Omar E. Pol, Mar 26 2021

a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/12 - 1/2 = 0.322467... . - Amiram Eldar, Nov 27 2023