A309288 - OEIS

%I #25 May 10 2023 04:30:53

%S 0,1,1,0,1,0,0,-1,1,1,1,0,-1,-2,-2,-1,3,2,2,1,0,1,1,0,-2,-2,-2,-2,-3,

%T -4,-4,-5,3,4,4,5,5,4,4,5,3,2,2,1,0,0,0,-1,-5,-5,-5,-4,-5,-6,-6,-5,-7,

%U -6,-6,-7,-6,-7,-7,-7,9,10,10,9,8,9,9,8,8,7,7,7

%N a(0) = 0, a(1) = 1, and for any n > 1, a(n) = Sum_{k > 1} (-1)^k * a(floor(n/k)).

%C This sequence is a signed variant of A022825.

%H Rémy Sigrist, <a href="/A309288/b309288.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f. A(x) satisfies -x * (1 - x) = Sum_{k>=1} (-1)^k * (1 - x^k) * A(x^k). - _Seiichi Manyama_, Mar 31 2023

%e a(3) = a(floor(3/2)) - a(floor(3/3)) = a(1) - a(1) = 0.

%t Join[{0}, Clear[a]; a[0]=0; a[1]=1; a[n_]:=a[n]=Sum[a[Floor[n/k]](-1)^k, {k, 2, n}]; Table[a[n], {n, 1, 100}]] (* _Vincenzo Librandi_, Jul 22 2019 *)

%t f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := If[e == 1, 0, 2^(e-2)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Join[{0}, Accumulate[Array[s, 100]]] (* _Amiram Eldar_, May 09 2023 *)

%o (PARI) a(n) = if (n<=1, n, sum (k=2, n, (-1)^k * a(n\k)))

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A309288(n):

%o     if n <= 1:

%o         return n

%o     c, j = 0, 2

%o     k1 = n//j

%o     while k1 > 1:

%o         j2 = n//k1 + 1

%o         c += ((j2-j)%2)*(1-2*(j%2))*A309288(k1)

%o         j, k1 = j2, n//j2

%o     return c+((n+1-j)%2)*(1-2*(j%2)) # _Chai Wah Wu_, Mar 31 2021

%Y Cf. A022825.

%K sign,look

%O 0,14

%A _Rémy Sigrist_, Jul 21 2019