A071323 Alternating sum of all divisors of n; divisors nondecreasing, starting with 1.

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

Offset: 1

Author

Reinhard Zumkeller, May 18 2002

Keywords

• sign

Examples

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Mihai Caragiu and Kaleb Swieringa, On the alternating sum-of-divisors, JP Journal of Algebra, Number Theory and Applications, 63(2) (2024), 97-110 (page 3, expression 4).

Crossrefs

Cf. A000203, A071321, A071324.

Programs

Mathematica

Table[Total[Times@@@Partition[Riffle[Divisors[n],{1,-1},{2,-1,2}],2]],{n,50}] (* Harvey P. Dale, Nov 05 2014 *)

Formula

a(n) = A071324(n) > 0 iff n square (iff A000005(n) is even), a(A000290(n)) > 0.

For primes p, a(p) = 1 - p.

From Shreyansh Jaiswal, Mar 04 2025: (Start)

a(n) = -A071324(n) < 0 iff n is nonsquare.

abs(a(n)) = A071324(n). (End)