A071324 - OEIS

%I #38 Feb 17 2024 18:23:16

%S 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,

%T 18,28,22,30,21,24,18,32,25,36,20,28,24,40,32,42,30,36,24,46,32,43,31,

%U 36,36,52,40,48,38,40,30,58,40,60,32,46,43,56,48,66,48,48,42,70,49,72

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

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

%C Alternating row sums of A056538. - _Omar E. Pol_, Feb 17 2024

%H Reinhard Zumkeller, <a href="/A071324/b071324.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

%F a(n) = Sum_{i=1..n} (A135539(n,i) mod 2). - _Ridouane Oudra_, Nov 23 2022

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

%p 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

%t a[n_] := Plus @@ (-(d = Divisors[n])*(-1)^(Range[Length[d],1,-1])); Array[a, 100] (* _Amiram Eldar_, Mar 11 2020 *)

%t Table[Total[Times@@@Partition[Riffle[Reverse[Divisors[n]],{1,-1},{2,-1,2}],2]],{n,80}] (* _Harvey P. Dale_, Nov 06 2022 *)

%o (PARI) a(n) = my(d=Vecrev(divisors(n))); sum(k=1, #d, (-1)^(k+1)*d[k]); \\ _Michel Marcus_, Aug 11 2018

%o (APL, Dyalog dialect)

%o divisors ← {⍺←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:⍺ ⋄ ⍺,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽⍺}

%o A071324 ← {-/⌽(divisors ⍵)} ⍝ _Antti Karttunen_, Feb 16 2024

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

%Y Cf. A134871, A135539.

%Y Cf. A056538.

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, May 18 2002, Jul 03 2008