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A304779 The "rootless" zeta function. Dirichlet inverse of the function defined by r(n) = (-1)^Omega(n) if n is 1 or not a perfect power and r(n) = 0 otherwise.

%I A304779 #15 Jul 28 2018 11:35:19
%S A304779 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,1,1,1,1,1,
%T A304779 1,5,1,1,1,2,1,1,1,2,2,1,1,3,1,2,1,2,1,2,1,2,1,1,1,4,1,1,2,1,1,1,1,2,
%U A304779 1,1,1,7,1,1,2,2,1,1,1,3,1,1,1,4,1,1,1,2,1,4,1,2,1,1,1,3,1,2,2,5,1,1,1,2,1
%N A304779 The "rootless" zeta function. Dirichlet inverse of the function defined by r(n) = (-1)^Omega(n) if n is 1 or not a perfect power and r(n) = 0 otherwise.
%C A304779 Omega(n) = A001222(n) is the number of prime factors of n counted with multiplicity.
%C A304779 First occurrence of k: 1, 12, 48, 60, 36, 3072, 72, 420, 240, 786432, 3145728, 144, 216, ..., . - _Robert G. Wilson v_, Jul 22 2018
%C A304779 Records: 1, 2, 5, 7, 12, 13, 15, 18, 26, 37, 38, 57, 60, 67, 81, 96, 142, 165, 199, 221, 234, ..., . - _Robert G. Wilson v_, Jul 22 2018
%H A304779 Antti Karttunen, <a href="/A304779/b304779.txt">Table of n, a(n) for n = 1..65537</a>
%F A304779 a(1) = 1 and a(n > 1) = -Sum_{d|n, d not a perfect power} (-1)^Omega(d) * a(n/d).
%t A304779 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 A304779 Array[a,100]
%o A304779 (PARI) A304779(n) = if(1==n,1,-sumdiv(n,d,if((d>1)&&!ispower(d),((-1)^bigomega(d))*A304779(n/d),0))); \\ _Antti Karttunen_, Jul 22 2018
%Y A304779 Positions of entries greater than 1 appear to be A126706.
%Y A304779 Cf. A000005, A000012, A000961, A001221, A001222, A001597, A005117, A007916, A008683, A008966, A091050, A303554, A304326, A304362, A304819.
%K A304779 nonn
%O A304779 1,12
%A A304779 _Gus Wiseman_, May 18 2018
%E A304779 More terms from _Antti Karttunen_, Jul 22 2018