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A365714 Sum of reversing binary value of n (A065620) and its Dirichlet inverse.

%I A365714 #8 Sep 19 2023 15:32:36
%S A365714 2,0,0,4,0,-4,0,8,1,-12,0,-4,0,12,6,16,0,-30,0,-12,-6,28,0,-8,9,20,15,
%T A365714 12,0,-32,0,32,-14,-60,-18,-28,0,60,-10,-24,0,64,0,28,61,-52,0,-16,9,
%U A365714 18,30,20,0,-66,-42,24,-30,-44,0,-20,0,44,-77,64,-30,-96,0,-60,26,160,0,-56,0,116,39,60,42,-96,0,-48,89
%N A365714 Sum of reversing binary value of n (A065620) and its Dirichlet inverse.
%H A365714 Antti Karttunen, <a href="/A365714/b365714.txt">Table of n, a(n) for n = 1..16384</a>
%F A365714 a(n) = A065620(n) + A365713(n).
%F A365714 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A065620(d) * A365713(n/d).
%F A365714 a(4*n)/4 = A065620(n).
%o A365714 (PARI)
%o A365714 A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<<i)); \\ From A065620 by _M. F. Hasler_
%o A365714 memoA365713 = Map();
%o A365714 A365713(n) = if(1==n,1,my(v); if(mapisdefined(memoA365713,n,&v), v, v = -sumdiv(n,d,if(d<n,A065620(n/d)*A365713(d),0)); mapput(memoA365713,n,v); (v)));
%o A365714 A365714(n) = (A065620(n)+A365713(n));
%Y A365714 Cf. A065620, A365713.
%Y A365714 Cf. also A365712.
%K A365714 sign
%O A365714 1,1
%A A365714 _Antti Karttunen_, Sep 19 2023