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%I #11 Nov 25 2021 12:16:01
%S 1,0,-1,-4,-11,-24,-57,-112,-243,-480,-1013,-1964,-4083,-8064,-16309,
%T -32496,-65519,-130440,-262125,-523156,-1048263,-2095104,-4194281,
%U -8383760,-16777015,-33546240,-67107609,-134200860,-268435427,-536835096,-1073741793,-2147417216,-4294962187,-8589803520,-17179867533,-34359463812
%N Dirichlet convolution of A000027 (identity function) with A349452 (Dirichlet inverse of A011782, 2^(n-1)).
%C Dirichlet convolution with A034729 gives sigma, A000203, and convolution with A034738 gives A018804.
%H Antti Karttunen, <a href="/A349569/b349569.txt">Table of n, a(n) for n = 1..1001</a>
%F a(n) = Sum_{d|n} d * A349452(n/d).
%t s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#]*2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, # * s[n/#] &]; Array[a, 36] (* _Amiram Eldar_, Nov 22 2021 *)
%o (PARI)
%o A011782(n) = (2^(n-1));
%o memoA349452 = Map();
%o A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(d<n,A011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
%o A349569(n) = sumdiv(n,d,d * A349452(n/d));
%Y Cf. A000027, A011782, A349452, A349570 (Dirichlet inverse).
%Y Cf. also A000203, A018804, A034729, A034738, A349563, A349565, A349567.
%K sign
%O 1,4
%A _Antti Karttunen_, Nov 22 2021
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