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A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

%I A354347 #17 Jun 09 2022 08:44:52
%S A354347 1,-1,-1,-1,-1,1,-1,-1,0,1,-1,1,-1,1,1,-9,-1,-2,-1,1,-3,1,-1,1,-4,-3,
%T A354347 0,1,-1,-1,-1,21,1,1,-1,-6,-1,1,1,3,-1,7,-1,-1,0,-3,-1,23,0,4,-3,7,-1,
%U A354347 2,1,3,1,1,-1,-1,-1,1,8,15,-1,-1,-1,1,1,3,-1,14,-1,1,-46,-7,-1,7,-1,5,0,1,-1,3,1,-3,1,-131
%N A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.
%H A354347 Antti Karttunen, <a href="/A354347/b354347.txt">Table of n, a(n) for n = 1..30030</a>
%H A354347 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A354347 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A345000(n/d) * a(d).
%F A354347 For all n >= 1, A000035(a(n)) = A353627(n).
%o A354347 (PARI)
%o A354347 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A354347 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A354347 A345000(n) = gcd(A003415(n),A003415(A276086(n)));
%o A354347 memoA354347 = Map();
%o A354347 A354347(n) = if(1==n,1,my(v); if(mapisdefined(memoA354347,n,&v), v, v = -sumdiv(n,d,if(d<n,A345000(n/d)*A354347(d),0)); mapput(memoA354347,n,v); (v)));
%Y A354347 Cf. A003415, A276086, A345000.
%Y A354347 Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).
%Y A354347 Cf. also A346242, A354348, A354823, A354824.
%K A354347 sign
%O A354347 1,16
%A A354347 _Antti Karttunen_, Jun 07 2022