A347030 - OEIS

%I #18 Apr 04 2023 17:05:08

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

%T -5,-6,10,11,10,11,11,10,9,10,6,5,6,5,3,3,2,1,-7,-7,-7,-6,-8,-9,-9,-8,

%U -12,-11,-12,-13,-11,-12,-13,-13,19,20,21,20,18,19,20,19,19,18,17,17

%N a(n) = 1 + Sum_{k=2..n} (-1)^k * a(floor(n/k)).

%C Partial sums of A067856.

%H Seiichi Manyama, <a href="/A347030/b347030.txt">Table of n, a(n) for n = 1..8191</a>

%H Ilya Gutkovskiy, <a href="/A347030/a347030.jpg">Scatterplot of a(n) up to n=10000</a>

%F G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x + Sum_{k>=2} (-1)^k * (1 - x^k) * A(x^k)).

%t a[n_] := a[n] = 1 + Sum[(-1)^k a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 75}]

%t nmax = 75; A[_] = 0; Do[A[x_] = (1/(1 - x)) (x + Sum[(-1)^k (1 - x^k) A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A347030(n):

%o     if n <= 1:

%o         return n

%o     c, j = 1, 2

%o     k1 = n//j

%o     while k1 > 1:

%o         j2 = n//k1 + 1

%o         c += (j2-j&1)*(-1 if j&1 else 1)*A347030(k1)

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

%o     return c+(n+1-j&1)*(-1 if j&1 else 1) # _Chai Wah Wu_, Apr 04 2023

%Y Cf. A002321, A025523, A067856, A309288, A347031.

%K sign

%O 1,2

%A _Ilya Gutkovskiy_, Aug 11 2021