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%I A323912 #17 Nov 22 2024 11:01:02
%S A323912 1,0,-1,0,-3,2,-5,0,-2,2,-9,4,-11,2,5,0,-15,2,-17,4,7,2,-21,8,-6,2,-4,
%T A323912 4,-27,-2,-29,0,11,2,17,8,-35,2,13,8,-39,-6,-41,4,8,2,-45,16,-10,-2,
%U A323912 17,4,-51,0,29,8,19,2,-57,4,-59,2,12,0,35,-14,-65,4,23,-10,-69,24,-71,2,4,4,47,-18,-77,16,-8,2,-81,-4,47,2,29,8,-87,4
%N A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.
%H A323912 Antti Karttunen, <a href="/A323912/b323912.txt">Table of n, a(n) for n = 1..20000</a>
%H A323912 Jon Maiga, <a href="http://sequencedb.net/s/A323912">Computer-generated formulas for A323912</a>, Sequence Machine.
%H A323912 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>.
%F A323912 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A083254(n/d) * a(d).
%F A323912 From _Antti Karttunen_, Nov 22 2024: (Start)
%F A323912 Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:
%F A323912 a(n) = Sum_{d|n} A323910(d).
%F A323912 a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).
%F A323912 a(n) = Sum_{d|n} A055615(d)*A067824(n/d)
%F A323912 a(n) = Sum_{d|n} A046692(d)*A253249(n/d)
%F A323912 a(n) = Sum_{d|n} A130054(d)*A174725(n/d)
%F A323912 a(n) = Sum_{d|n} A101035(d)*A330575(n/d)
%F A323912 a(n) = Sum_{d|n} A191161(d)*A328722(n/d)
%F A323912 (End)
%o A323912 (PARI)
%o A323912 up_to = 16384;
%o A323912 DirInverse(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 A323912 A083254(n) = (2*eulerphi(n)-n);
%o A323912 v323912 = DirInverse(vector(up_to,n,A083254(n)));
%o A323912 A323912(n) = v323912[n];
%o A323912 (PARI)
%o A323912 A083254(n) = (2*eulerphi(n)-n);
%o A323912 memoA323912 = Map();
%o A323912 A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(d<n,A083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ _Antti Karttunen_, Nov 22 2024
%Y A323912 Cf. A083254, A323913.
%Y A323912 Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.
%K A323912 sign
%O A323912 1,5
%A A323912 _Antti Karttunen_, Feb 12 2019