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A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

%I A297114 #29 Dec 16 2024 13:29:34
%S A297114 0,0,0,0,2,-2,3,0,3,-2,7,-4,9,-2,0,0,14,-6,15,-4,4,-2,18,-8,14,-2,7,
%T A297114 -4,24,-14,25,0,9,-2,14,-12,33,-2,9,-8,37,-18,38,-4,3,-2,41,-16,35,
%U A297114 -10,12,-4,48,-18,24,-8,15,-2,53,-28,55,-2,6,0,33,-26,63,-4,21,-22,66,-24,69,-2,6,-4,44,-30,73,-16,28,-2
%N A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.
%H A297114 Antti Karttunen, <a href="/A297114/b297114.txt">Table of n, a(n) for n = 1..16384</a>
%H A297114 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A297114 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A297114 a(n) = Sum_{d|n} A008683(n/d)*A294898(d).
%F A297114 a(n) = A297111(n) - n.
%F A297114 a(n) = A297117(n) - A051953(n).
%F A297114 a(n) = A083254(n) - A297115(n).
%F A297114 a(2n) = A083254(2n) = A378986(n) = -2*A176095(n).
%F A297114 a(n) = A294898(n) - A317844(n).
%t A297114 Table[DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &], {n, 82}] (* _Michael De Vlieger_, Mar 11 2019 *)
%o A297114 (PARI)
%o A297114 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A297114 A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));
%Y A297114 Cf. A000203, A005187, A051953, A083254, A294898, A297111, A297115, A297117, A317844, A324397.
%Y A297114 Cf. A378986 (even bisection, -2*A176095).
%K A297114 sign
%O A297114 1,5
%A A297114 _Antti Karttunen_, Dec 26 2017