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A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

%I A346242 #26 Jun 09 2022 08:44:47
%S A346242 1,-1,-3,0,-1,5,-1,0,6,-3,-1,-2,-1,1,-9,0,-1,-16,-1,4,3,1,-1,0,-24,1,
%T A346242 -12,0,-1,43,-1,0,3,1,-5,14,-1,1,3,0,-1,-11,-1,0,54,1,-1,0,-6,32,3,0,
%U A346242 -1,44,-3,-6,3,1,-1,-50,-1,1,-24,0,1,-5,-1,0,3,-15,-1,-4,-1,1,96,0,-5,-5,-1,0,24,1,-1,8,-3,1,3,0,-1
%N A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).
%H A346242 Antti Karttunen, <a href="/A346242/b346242.txt">Table of n, a(n) for n = 1..30030</a>
%H A346242 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A346242 a(n) = A346243(n) - A324198(n).
%F A346242 From _Antti Karttunen_, Jun 09 2022: (Start)
%F A346242 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A324198(n/d) * a(d).
%F A346242 For all n >= 1, A000035(a(n)) = A008966(n).
%F A346242 For all n >= 1, a(A045344(n)) = -1.
%F A346242 (End)
%o A346242 (PARI)
%o A346242 up_to = 65537;
%o A346242 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A346242 A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
%o A346242 v346242 = DirInverseCorrect(vector(up_to,n,A324198(n)));
%o A346242 A346242(n) = v346242[n];
%Y A346242 Cf. A276086, A324198, A346243.
%Y A346242 Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).
%Y A346242 Cf. also A354347, A354348, A354823, A354824.
%K A346242 sign,base
%O A346242 1,3
%A A346242 _Antti Karttunen_, Jul 13 2021