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A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

%I A349570 #14 Jan 09 2024 13:19:42
%S A349570 1,0,1,4,11,24,57,112,244,480,1013,1972,4083,8064,16331,32512,65519,
%T A349570 130488,262125,523244,1048377,2095104,4194281,8384176,16777136,
%U A349570 33546240,67108096,134201316,268435427,536836584,1073741793,2147418112,4294964213,8589803520,17179868787,34359470272,68719476699,137438429184,274877894643
%N A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).
%C A349570 Dirichlet convolution of this sequence with phi (A000010) is A000740, with sigma (A000203) it is A034729, and with A018804 it is A034738.
%H A349570 Antti Karttunen, <a href="/A349570/b349570.txt">Table of n, a(n) for n = 1..1001</a>
%F A349570 a(n) = Sum_{d|n} 2^(d-1) * A055615(n/d).
%t A349570 a[n_] := DivisorSum[n, # * MoebiusMu[#] * 2^(n/# - 1) &]; Array[a, 40] (* _Amiram Eldar_, Nov 22 2021 *)
%o A349570 (PARI)
%o A349570 A055615(n) = (n*moebius(n));
%o A349570 A349570(n) = sumdiv(n,d,(2^(d-1)) * A055615(n/d));
%Y A349570 Cf. A011782, A055615, A349569 (Dirichlet inverse).
%Y A349570 Cf. also A000010, A000740, A000203, A018804, A034729, A034738, A349564, A349566, A349568.
%K A349570 nonn
%O A349570 1,4
%A A349570 _Antti Karttunen_, Nov 22 2021