A071323 -id:A071323 - cp's OEIS frontend

A071321 Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n).

0, 2, 3, 0, 5, -1, 7, 2, 0, -3, 11, 3, 13, -5, -2, 0, 17, 2, 19, 5, -4, -9, 23, -1, 0, -11, 3, 7, 29, 4, 31, 2, -8, -15, -2, 0, 37, -17, -10, -3, 41, 6, 43, 11, 5, -21, 47, 3, 0, 2, -14, 13, 53, -1, -6, -5, -16, -27, 59, -2, 61, -29, 7, 0, -8

Offset: 1

Reinhard Zumkeller, May 18 2002

sign

a(n) = 0 iff n square, a(A000290(n)) = 0;
a(n) <= 0 iff A001222(n) is even;
a(n) = n iff n prime, a(A000040(n)) = A000040(n).
a(2n) = -a(n) + 2. - Ralf Stephan

			72 = 2*2*2*3*3, therefore a(72) = 2 - 2 + 2 - 3 + 3 = 2;
90 = 2*3*3*5, therefore a(90) = 2 - 3 + 3 - 5 = -3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001414, A020639, A071322, A071323.
Cf. A008836.
Cf. A027746, A033999.

a071321 1 = 0
a071321 n = sum $ zipWith (*) a033999_list $ a027746_row n
-- Reinhard Zumkeller, Jun 01 2013

Join[{0},Table[Total[Times@@@Partition[Riffle[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[n]],{1,-1},{2,-1,2}],2]],{n,2,100}]] (* Harvey P. Dale, Sep 23 2015 *)

from sympy import factorint
def A071321(n):
    fs = factorint(n,multiple=True)
    return sum(fs[::2])-sum(fs[1::2]) # Chai Wah Wu, Aug 23 2021

a(n) = -A071322(n)*A008836(n). - Franklin T. Adams-Watters, Oct 18 2006

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

Original entry on oeis.org

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

Reinhard Zumkeller, May 18 2002, Jul 03 2008

nonn

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

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

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

Cf. A000010, A000203, A071322, a(n) = abs(A071323(n)).
Cf. A134871, A135539.
Cf. A056538.

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

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 *)

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

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

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)

A370681 a(n) is the alternating sum of the unitary divisors of n, when these divisors are starting with n and decreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 7, 8, 6, 10, 10, 12, 8, 12, 15, 16, 10, 18, 18, 16, 12, 22, 18, 24, 14, 26, 24, 28, 22, 30, 31, 24, 18, 32, 30, 36, 20, 28, 36, 40, 32, 42, 36, 40, 24, 46, 34, 48, 26, 36, 42, 52, 28, 48, 54, 40, 30, 58, 46, 60, 32, 60, 63, 56, 48, 66, 54

Offset: 1

Amiram Eldar, Feb 26 2024

a(n) is odd if and only if n is a power of 2.

			The unitary divisors of 6 are {1, 2, 3, 6}, hence a(6) = 6 - 3 + 2 - 1 = 4.
The unitary divisors of 12 are {1, 3, 4, 12}, hence a(12) = 12 - 4 + 3 - 1 = 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A034448, A071323, A071324, A077610, A370683.

a[n_] := Module[{d = Reverse[Select[Divisors[n], CoprimeQ[#, n/#] &]]}, Total[(-1)^(Range[Length[d]] + 1)*d]]; Array[a, 100]

a(n) = {my(d = Vecrev(select(x->(gcd(x, n/x) == 1), divisors(n)))); sum(i=1, #d, (-1)^(i+1)*d[i]);}

a(n) = A071324(n) if n is squarefree (A005117) or if n is in A370683.

cp's OEIS Frontend

A071321 Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n).

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

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A370681 a(n) is the alternating sum of the unitary divisors of n, when these divisors are starting with n and decreasing.

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Mathematica

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