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A345065 Sum of A011772 and its Dirichlet inverse.

%I A345065 #14 Jul 05 2021 02:43:41
%S A345065 2,0,0,9,0,12,0,15,4,24,0,-8,0,36,16,31,0,24,0,-28,24,60,0,33,16,72,
%T A345065 24,-36,0,-74,0,63,40,96,48,9,0,108,48,86,0,-116,0,-64,36,132,0,14,36,
%U A345065 32,64,-84,0,8,80,23,72,168,0,341,0,180,48,127,96,-190,0,-112,88,-244,0,-77,0,216,40,-120,120,-240,0,-73,88
%N A345065 Sum of A011772 and its Dirichlet inverse.
%H A345065 Antti Karttunen, <a href="/A345065/b345065.txt">Table of n, a(n) for n = 1..10000</a>
%H A345065 Antti Karttunen, <a href="/A345065/a345065.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A345065 a(n) = A011772(n) + A345055(n).
%F A345065 a(2^i) = 2^(i+1)-1 for i >= 3. See A345053. - _Chai Wah Wu_, Jul 05 2021
%o A345065 (PARI)
%o A345065 up_to = 65537;
%o A345065 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 (correctly!)
%o A345065 A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
%o A345065 v345055 = DirInverseCorrect(vector(up_to,n,A011772(n)));
%o A345065 A345055(n) = v345055[n];
%o A345065 A345065(n) = (A011772(n)+A345055(n));
%Y A345065 Cf. A011772, A345053, A345055.
%K A345065 sign
%O A345065 1,1
%A A345065 _Antti Karttunen_, Jun 20 2021