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

1,3

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).

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)

Example

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

Mathematica

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

Crossrefs

Cf. A000203, A071321, A071324.

Sequence in context: A370829 A228286 A158973 * A071324 A361003 A321441

Adjacent sequences: A071320 A071321 A071322 * A071324 A071325 A071326

Keyword

sign

Author

Reinhard Zumkeller, May 18 2002

Status

approved