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A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).

%I A349373 #10 Nov 21 2021 10:17:15
%S A349373 1,0,0,-1,-1,0,-2,-2,-1,0,-4,0,-5,0,2,-3,-7,0,-8,1,3,0,-10,0,-3,0,-2,
%T A349373 2,-13,0,-14,-4,5,0,8,1,-17,0,6,2,-19,0,-20,4,5,0,-22,0,-5,0,8,5,-25,
%U A349373 0,14,4,9,0,-28,-2,-29,0,8,-5,17,0,-32,7,11,0,-34,2,-35,0,4,8,23,0,-38,3,-3,0,-40,-3,23,0,14,8
%N A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).
%H A349373 Antti Karttunen, <a href="/A349373/b349373.txt">Table of n, a(n) for n = 1..20000</a>
%F A349373 a(n) = Sum_{d|n} A003602(n/d) * A023900(d).
%t A349373 f[p_, e_] := (1 - p); d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * d[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 16 2021 *)
%o A349373 (PARI)
%o A349373 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349373 A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
%o A349373 A349373(n) = sumdiv(n,d,A003602(n/d)*A023900(d));
%Y A349373 Cf. A003602, A023900.
%Y A349373 Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.
%K A349373 sign
%O A349373 1,7
%A A349373 _Antti Karttunen_, Nov 15 2021