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
Keywords
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
Programs
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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
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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 *)
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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
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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(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)
Comments