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A354866 Dirichlet inverse of A122111.

%I A354866 #13 Jun 10 2022 11:09:15
%S A354866 1,-2,-4,1,-8,10,-16,-1,7,20,-32,-10,-64,40,46,2,-128,-27,-256,-20,92,
%T A354866 80,-512,14,37,160,-17,-40,-1024,-150,-2048,-3,184,320,202,53,-4096,
%U A354866 640,368,28,-8192,-300,-16384,-80,-146,1280,-32768,-26,175,-129,736,-160,-65536,85,404,56,1472,2560,-131072,242,-262144
%N A354866 Dirichlet inverse of A122111.
%H A354866 Antti Karttunen, <a href="/A354866/b354866.txt">Table of n, a(n) for n = 1..10000</a>
%H A354866 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A354866 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A122111(n/d) * a(d).
%F A354866 a(n) = A354867(n) - A122111(n).
%o A354866 (PARI)
%o A354866 A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
%o A354866 memoA354866 = Map();
%o A354866 A354866(n) = if(1==n,1,my(v); if(mapisdefined(memoA354866,n,&v), v, v = -sumdiv(n,d,if(d<n,A122111(n/d)*A354866(d),0)); mapput(memoA354866,n,v); (v)));
%Y A354866 Cf. A122111, A354867, A354868 (parity), A354869 (positions of odd terms).
%Y A354866 Cf. also A354186, A354349, A354351, A354359, A354826.
%K A354866 sign
%O A354866 1,2
%A A354866 _Antti Karttunen_, Jun 09 2022