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A324640 Dirichlet inverse of the Doudna sequence, A005940.

%I A324640 #10 Mar 11 2019 20:47:22
%S A324640 1,-2,-3,0,-5,6,-9,0,2,10,-15,0,-25,18,3,0,-11,-4,-21,0,19,30,-45,0,
%T A324640 -24,50,-60,0,-125,-6,-81,0,77,22,57,0,-55,42,87,0,-77,-38,-105,0,-78,
%U A324640 90,-135,0,-40,48,-81,0,-245,120,-75,0,-217,250,-375,0,-625,162,-150,0,233,-154,-39,0,205,-114,-99,0,-91,110,174,0,-5,-174,-189,0
%N A324640 Dirichlet inverse of the Doudna sequence, A005940.
%H A324640 Antti Karttunen, <a href="/A324640/b324640.txt">Table of n, a(n) for n = 1..16384</a>
%F A324640 a(1) = 1; for n > 1, a(n) = -Sum_{d|n, d<n} a(d) * A005940(n/d).
%F A324640 a(p) = -A005940(p) for all primes p.
%o A324640 (PARI)
%o A324640 up_to = 16384;
%o A324640 DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -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 A324640 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
%o A324640 v324640 = DirInverse(vector(up_to,n,A005940(n)));
%o A324640 A324640(n) = v324640[n];
%Y A324640 Cf. A005940, A324106, A324641.
%K A324640 sign
%O A324640 1,2
%A A324640 _Antti Karttunen_, Mar 11 2019