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

1,3

Comments

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

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

Links

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

Formula

Equals A054525 * A134871; i.e., Mobius transform of [1, 2, 3, 5, 5, 8, 7, 10, 10, 12, 11, ...]. - Gary W. Adamson, Nov 14 2007

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A134871, A135539.

Cf. A056538.

Sequence in context: A228286 A158973 A071323 * A361003 A321441 A063655

Adjacent sequences: A071321 A071322 A071323 * A071325 A071326 A071327

Keyword

nonn

Author

Reinhard Zumkeller, May 18 2002, Jul 03 2008

Status

approved