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A071324
Alternating sum of all divisors of n; divisors nonincreasing, starting with n.
24
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
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. A056538.
Sequence in context: A228286 A158973 A071323 * A361003 A321441 A063655
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, May 18 2002, Jul 03 2008
STATUS
approved