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A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

%I A349431 #19 Jun 02 2025 15:24:52
%S A349431 1,-1,-1,-1,-2,1,-3,-1,-1,2,-5,1,-6,3,4,-1,-8,1,-9,2,6,5,-11,1,-2,6,
%T A349431 -1,3,-14,-4,-15,-1,10,8,12,1,-18,9,12,2,-20,-6,-21,5,4,11,-23,1,-3,2,
%U A349431 16,6,-26,1,20,3,18,14,-29,-4,-30,15,6,-1,24,-10,-33,8,22,-12,-35,1,-36,18,4,9,30,-12,-39,2,-1,20
%N A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).
%C A349431 Dirichlet convolution of this sequence with A000010 gives A349136, which also proves the formula involving A023900.
%C A349431 Convolution with A000203 gives A349371.
%H A349431 Antti Karttunen, <a href="/A349431/b349431.txt">Table of n, a(n) for n = 1..20000</a>
%F A349431 a(n) = Sum_{d|n} A003602(n/d) * A055615(d).
%F A349431 a(n) = A023900(n) when n is a power of 2, and a(n) = A023900(n)/2 for all other numbers.
%F A349431 a(n) = -A297381(n).
%t A349431 k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, # * MoebiusMu [#] * k[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349431 (PARI)
%o A349431 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349431 A055615(n) = (n*moebius(n));
%o A349431 A349431(n) = sumdiv(n,d,A003602(n/d)*A055615(d));
%o A349431 (PARI)
%o A349431 A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
%o A349431 A349431(n) = if(!bitand(n,n-1),A023900(n),A023900(n)/2);
%Y A349431 Sequence A297381 negated.
%Y A349431 Cf. A003602, A023900, A055615, A297381, A349432 (Dirichlet inverse), A349433 (sum with it).
%Y A349431 Cf. also A000010, A000203, A349136, A349371, and also A349444, A349447.
%K A349431 sign
%O A349431 1,5
%A A349431 _Antti Karttunen_, Nov 17 2021