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A300723 Möbius-transform of A005187(A053645(n)).

%I A300723 #8 Mar 24 2018 18:54:00
%S A300723 0,0,1,0,1,2,4,0,0,2,4,4,8,6,9,0,1,0,4,4,3,6,11,8,15,10,18,12,23,10,
%T A300723 26,0,-4,2,-1,0,8,6,2,8,16,2,19,12,12,14,26,16,28,16,33,20,39,20,37,
%U A300723 24,42,26,50,20,54,30,49,0,-8,-6,4,4,-4,-2,11,0,16,10,-7,12,15,2,26,16,13,18,35,4,37,22,18,24,47,12
%N A300723 Möbius-transform of A005187(A053645(n)).
%H A300723 Antti Karttunen, <a href="/A300723/b300723.txt">Table of n, a(n) for n = 1..16384</a>
%F A300723 a(n) = Sum_{d|n} A008683(n/d)*A005187(A053645(d)).
%F A300723 a(1) = 0; for n > 1, a(n) = A297111(n) - 2*A300724(n).
%t A300723 With[{s = Array[2 # - DigitCount[2 #, 2, 1] &[# - 2^Floor@ Log2@ #] &, 90]}, Table[DivisorSum[n, MoebiusMu[n/#] s[[#]] &], {n, Length@ s}]] (* _Michael De Vlieger_, Mar 13 2018 *)
%o A300723 (PARI)
%o A300723 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A300723 A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
%o A300723 A053645(n) = (n-A053644(n));
%o A300723 A300723(n) = sumdiv(n,d,moebius(n/d)*A005187(A053645(d)));
%Y A300723 Cf. A005187, A008683, A053645, A297111, A300724, A300725.
%K A300723 sign
%O A300723 1,6
%A A300723 _Antti Karttunen_, Mar 12 2018