A378989 - OEIS

%I #7 Dec 15 2024 09:30:27

%S 1,0,-1,0,-1,0,-2,0,1,0,-2,0,-2,0,1,0,-1,0,-2,0,5,0,-3,0,0,0,-3,0,-3,

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

%U 0,-4,0,-11,0,6,0,-2,0,8,0,-3,0,-2,0,2,0,9,0,-4,0,6,0,-3,0,1,0,6,0,-3,0,8,0,9,0,2,0,-2,0,-12,0,-3,0,-4,0,-12

%N Dirichlet inverse of the Möbius transform of binary weight of n.

%H Antti Karttunen, <a href="/A378989/b378989.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A297115(n/d) * a(d).

%F a(n) = Sum_{d|n} A378990(d).

%o (PARI)

%o A297115(n) = sumdiv(n, d, moebius(n/d)*hammingweight(d));

%o memoA378989 = Map();

%o A378989(n) = if(1==n,1,my(v); if(mapisdefined(memoA378989,n,&v), v, v = -sumdiv(n,d,if(d<n,A297115(n/d)*A378989(d),0)); mapput(memoA378989,n,v); (v)));

%Y Dirichlet inverse of A297115.

%Y Inverse Möbius transform of A378990.

%Y Cf. A000120.

%K sign

%O 1,7

%A _Antti Karttunen_, Dec 15 2024