cp's OEIS Frontend

A092149 Partial sums of A092673.

%I A092149 #28 Mar 31 2023 13:22:14
%S A092149 1,-1,-2,-1,-2,0,-1,-1,-1,1,0,-1,-2,0,1,1,0,0,-1,-2,-1,1,0,0,0,2,2,1,
%T A092149 0,-2,-3,-3,-2,0,1,1,0,2,3,3,2,0,-1,-2,-2,0,-1,-1,-1,-1,0,-1,-2,-2,-1,
%U A092149 -1,0,2,1,2,1,3,3,3,4,2,1,0,1,-1,-2,-2,-3,-1,-1,-2,-1,-3,-4,-4,-4,-2,-3,-2,-1,1,2,2,1,1,2,1,2,4,5,5,4,4,4,4,3,1,0,0,-1,1
%N A092149 Partial sums of A092673.
%H A092149 Michael De Vlieger, <a href="/A092149/b092149.txt">Table of n, a(n) for n = 1..10000</a>
%F A092149 G.f. Sum_{n >= 1} a(n)*(x^n)/((1-x^n)*(x^(n+1)-1))*x = -(x^2) and -1/x. [_Mats Granvik_, Oct 11 2010]
%F A092149 On the Riemann hypothesis, |a(n)| = O(n^(1/2+e)) for any e > 0. - _Charles R Greathouse IV_, Feb 07 2013
%F A092149 a(1)=1, then for n>=2, Sum_{k=1..n} a(floor(n/k)) = 0. - _Benoit Cloitre_, Feb 21 2013
%F A092149 G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} (1 - x^k) * A(x^k). - _Seiichi Manyama_, Mar 31 2023
%t A092149 Accumulate@ Array[MoebiusMu[#] - If[OddQ@ #, 0, MoebiusMu[#/2]] &, 106] (* _Michael De Vlieger_, Mar 31 2021 *)
%o A092149 (PARI) a(n)=my(s);forstep(k=bitor(n\4+1,1),n\2,2,s-=moebius(k));forstep(k=bitor(n\2+1,1),n,2,s+=moebius(k)); s \\ _Charles R Greathouse IV_, Feb 07 2013
%o A092149 (Python)
%o A092149 from functools import lru_cache
%o A092149 @lru_cache(maxsize=None)
%o A092149 def A092149(n):
%o A092149     if n == 1:
%o A092149         return 1
%o A092149     c, j = n+1, 2
%o A092149     k1 = n//j
%o A092149     while k1 > 1:
%o A092149         j2 = n//k1 + 1
%o A092149         c += (j2-j)*A092149(k1)
%o A092149         j, k1 = j2, n//j2
%o A092149     return j-c # _Chai Wah Wu_, Mar 31 2021
%Y A092149 Cf. A022825, A092673, A309288.
%K A092149 sign,look
%O A092149 1,3
%A A092149 _Jon Perry_, Mar 31 2004