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A346477 Dirichlet inverse of A346476.

%I A346477 #20 Dec 23 2024 22:49:00
%S A346477 1,-1,-1,2,-3,5,-3,2,8,13,-9,-2,-9,17,11,8,-15,-8,-15,-12,19,37,-17,
%T A346477 18,8,41,-4,-12,-27,-33,-25,20,37,61,25,56,-33,65,35,38,-39,-45,-39,
%U A346477 -42,-36,77,-41,32,32,-20,53,-42,-47,96,35,58,61,109,-57,132,-55,109,-48,56,43,-121,-63,-72,71,-109,-69,56
%N A346477 Dirichlet inverse of A346476.
%H A346477 Antti Karttunen, <a href="/A346477/b346477.txt">Table of n, a(n) for n = 1..10000</a>
%H A346477 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A346477 a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, d<n} a(d) * A346476(n/d).
%F A346477 a(n) = A346478(n) - A346476(n).
%F A346477 a(p) = A252748(p) = A346248(p) = -A346476(p) = -A062234(A000720(p)), for any prime p.
%o A346477 (PARI)
%o A346477 up_to = 16384;
%o A346477 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 A346477 A346476(n) = (n+n-A250469(n));
%o A346477 v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
%o A346477 A346477(n) = v346477[n];
%Y A346477 Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346478.
%Y A346477 Cf. also A323910, A346248, A346479.
%K A346477 sign
%O A346477 1,4
%A A346477 _Antti Karttunen_, Jul 29 2021