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A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

%I A349442 #8 Nov 26 2021 16:35:25
%S A349442 1,1,2,-1,4,2,6,-3,-2,4,10,-2,12,6,8,-1,16,-2,18,-4,12,10,22,-6,-4,12,
%T A349442 -16,-6,28,8,30,5,20,16,24,2,36,18,24,-12,40,12,42,-10,-8,22,46,-2,-6,
%U A349442 -4,32,-12,52,-16,40,-18,36,28,58,-8,60,30,-12,7,48,20,66,-16,44,24,70,6,72,36,-8,-18,60,24,78,-4
%N A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).
%C A349442 Multiplicative because both A000027 and A349350 are.
%H A349442 Antti Karttunen, <a href="/A349442/b349442.txt">Table of n, a(n) for n = 1..20000</a>
%F A349442 a(n) = Sum_{d|n} d * A349350(n/d).
%o A349442 (PARI)
%o A349442 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 A349442 memoA349350 = Map();
%o A349442 A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(d<n,A057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));
%o A349442 A349442(n) = sumdiv(n,d,d*A349350(n/d));
%Y A349442 Cf. A000027, A057521, A349350, A349441 (Dirichlet inverse), A349443 (sum with it).
%K A349442 sign,mult
%O A349442 1,3
%A A349442 _Antti Karttunen_, Nov 18 2021