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A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

%I A349441 #14 Nov 27 2021 11:04:22
%S A349441 1,-1,-2,2,-4,2,-6,0,6,4,-10,-4,-12,6,8,0,-16,-6,-18,-8,12,10,-22,0,
%T A349441 20,12,0,-12,-28,-8,-30,0,20,16,24,12,-36,18,24,0,-40,-12,-42,-20,-24,
%U A349441 22,-46,0,42,-20,32,-24,-52,0,40,0,36,28,-58,16,-60,30,-36,0,48,-20,-66,-32,44,-24,-70,0,-72,36,-40,-36
%N A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).
%C A349441 Multiplicative because A055615 and A057521 are.
%C A349441 Convolving this with Euler phi (A000010) produces A349379.
%H A349441 Antti Karttunen, <a href="/A349441/b349441.txt">Table of n, a(n) for n = 1..20000</a>
%F A349441 a(n) = Sum_{d|n} A057521(n/d) * A055615(d).
%F A349441 Multiplicative with a(p^e) = 1 - p is e = 1, p^2 - p if e = 2, and 0 otherwise. - _Amiram Eldar_, Nov 19 2021
%t A349441 f[p_, e_] := Which[e > 2, 0, e == 2, p^2 - p, e == 1, 1 - p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)
%o A349441 (PARI)
%o A349441 A055615(n) = (n*moebius(n));
%o A349441 A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
%o A349441 A349441(n) = sumdiv(n,d,A057521(n/d)*A055615(d));
%Y A349441 Cf. A055615, A057521, A349442 (Dirichlet inverse), A349443 (sum with it).
%Y A349441 Cf. also A097945, A349379.
%K A349441 sign,mult,look
%O A349441 1,3
%A A349441 _Antti Karttunen_, Nov 18 2021