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A304820 A co-delta function for non-perfect powers. Dirichlet inverse of A304819.

%I A304820 #13 Aug 23 2018 02:21:21
%S A304820 1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A304820 0,2,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,
%U A304820 0,0,0,2,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,1,2,0,0,0,0,0
%N A304820 A co-delta function for non-perfect powers. Dirichlet inverse of A304819.
%H A304820 Antti Karttunen, <a href="/A304820/b304820.txt">Table of n, a(n) for n = 1..65537</a>
%F A304820 a(n) = Sum_{d|n} A304779(d) * mu(n/d), where A304779 is the Dirichlet inverse of A304653.
%t A304820 a[n_]:=a[n]=If[n==1,1,-Sum[(-1)^PrimeOmega[d]*a[n/d],{d,Select[Rest[Divisors[n]],GCD@@FactorInteger[#][[All,2]]==1&]}]];
%t A304820 Table[Sum[a[d]*MoebiusMu[n/d],{d,Divisors[n]}],{n,100}]
%o A304820 (PARI)
%o A304820 A304819(n) = sumdiv(n,d,if(!ispower(d),(-1)^bigomega(d),0));
%o A304820 A304820(n) = if(1==n,1,-sumdiv(n,d,if(d<n,A304819(n/d)*A304820(d),0))); \\ _Antti Karttunen_, Jul 29 2018
%o A304820 (PARI)
%o A304820 A304779(n) = if(1==n,1,-sumdiv(n,d,if((d>1)&&!ispower(d),((-1)^bigomega(d))*A304779(n/d),0)));
%o A304820 A304820(n) = sumdiv(n,d,moebius(n/d)*A304779(d)); \\ _Antti Karttunen_, Jul 29 2018
%Y A304820 Positions of nonzero entries appear to be A126706.
%Y A304820 Cf. A000005, A000007, A001222, A001597, A005117, A007916, A008683, A008836, A304362, A304653, A304779, A304817, A304819.
%K A304820 nonn
%O A304820 1,36
%A A304820 _Gus Wiseman_, May 19 2018
%E A304820 More terms from _Antti Karttunen_, Jul 29 2018