A071324 Alternating sum of all divisors of n; divisors nonincreasing, starting with n.

1, 1, 2, 3, 4, 4, 6, 5, 7, 6, 10, 8, 12, 8, 12, 11, 16, 13, 18, 12, 16, 12, 22, 16, 21, 14, 20, 18, 28, 22, 30, 21, 24, 18, 32, 25, 36, 20, 28, 24, 40, 32, 42, 30, 36, 24, 46, 32, 43, 31, 36, 36, 52, 40, 48, 38, 40, 30, 58, 40, 60, 32, 46, 43, 56, 48, 66, 48, 48, 42, 70, 49, 72

Offset: 1

Author

Reinhard Zumkeller, May 18 2002, Jul 03 2008

Keywords

• nonn

Comments

Alternating row sums of A056538. - Omar E. Pol, Feb 17 2024

Does a constant analogous to the median abundancy index (see A353617) exist for this function? Particularly, does a constant exist such that the numbers having the value a(n)/n greater than this constant have natural density exactly 1/2? Using the first 10^7 values, one observes that if it exists, this constant appears to converge around 0.726. - Shreyansh Jaiswal, Apr 16 2025

Examples

			Divisors of 20: {1,2,4,5,10,20} therefore a(20) = 20 - 10 + 5 - 4 + 2 - 1 = 12.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Krassimir Atanassov, A remark on an arithmetic function. Part 2, Notes on Number Theory and Discrete Mathematics 15(3) (2009), 21-22.
• Shreyansh Jaiswal, On two Conjectures by Atanassov on the Alternating sum-of-divisors function, Jun 11 2025

Crossrefs

Cf. A000010, A000203, A071322, a(n) = abs(A071323(n)).

Cf. A134871, A135539.

Cf. A056538.

Programs

Maple

with(numtheory): a:=proc(n) local k, t:=0, A:=divisors(n); for k to tau(n) do t:= t+A[k]*(-1)^(tau(n)-k) end do; return t; end proc; seq(a(n), n=1..60); # Ridouane Oudra, Nov 23 2022

Mathematica

a[n_] := Plus @@ (-(d = Divisors[n])*(-1)^(Range[Length[d],1,-1])); Array[a, 100] (* Amiram Eldar, Mar 11 2020 *)
Table[Total[Times@@@Partition[Riffle[Reverse[Divisors[n]],{1,-1},{2,-1,2}],2]],{n,80}] (* Harvey P. Dale, Nov 06 2022 *)

PARI

a(n) = my(d=Vecrev(divisors(n))); sum(k=1, #d, (-1)^(k+1)*d[k]); \\ Michel Marcus, Aug 11 2018
(APL, Dyalog dialect)
divisors ← {⍺←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:⍺ ⋄ ⍺,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽⍺}
A071324 ← {-/⌽(divisors ⍵)} ⍝ Antti Karttunen, Feb 16 2024

Python

from sympy import divisors;  from functools import lru_cache
cached_divisors = lru_cache()(divisors)
def a(n):  return sum(d if i%2==0 else -d for i, d in enumerate(reversed(cached_divisors(n))))
A071324 = [a(i) for i in range(1, 74)]  # Jwalin Bhatt, Apr 02 2025

Formula

a(A028983(n)) mod 2 = 0; a(A028982(n)) mod 2 = 1.

a(n) = Sum_{i=1..n} (A135539(n,i) mod 2). - Ridouane Oudra, Nov 23 2022

From Shreyansh Jaiswal, Apr 16 2025: (Start)

a(p) = p-1 for prime p.

For odd n, 2n/3 <= a(n) <= n.

For even n, n/2 <= a(n) <= 5n/6.

a(n) >= A000010(n) for n>=1. (End)