cp's OEIS Frontend

A346241 Dirichlet inverse of pointwise sum of A003415 (arithmetic derivative of n) and A063524 (1, 0, 0, 0, ...).

%I A346241 #20 Jun 08 2022 15:48:43
%S A346241 1,-1,-1,-3,-1,-3,-1,-5,-5,-5,-1,-1,-1,-7,-6,-3,-1,-2,-1,-5,-8,-11,-1,
%T A346241 17,-9,-13,-16,-9,-1,3,-1,11,-12,-17,-10,33,-1,-19,-14,19,-1,1,-1,-17,
%U A346241 -14,-23,-1,63,-13,-14,-18,-21,-1,28,-14,21,-20,-29,-1,76,-1,-31,-22,45,-16,-3,-1,-29,-24,-9,-1,112,-1,-37,-22,-33
%N A346241 Dirichlet inverse of pointwise sum of A003415 (arithmetic derivative of n) and A063524 (1, 0, 0, 0, ...).
%H A346241 Antti Karttunen, <a href="/A346241/b346241.txt">Table of n, a(n) for n = 1..10000</a>
%H A346241 Antti Karttunen, <a href="/A346241/a346241.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A346241 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A003415(n/d) * a(d).
%o A346241 (PARI)
%o A346241 up_to = 65537;
%o A346241 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 A346241 A003415plusA063524(n) = if(n<=1, 1, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A346241 v346241 = DirInverseCorrect(vector(up_to,n,A003415plusA063524(n)));
%o A346241 A346241(n) = v346241[n];
%o A346241 (PARI)
%o A346241 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A346241 memoA346241 = Map();
%o A346241 A346241(n) = if(1==n,1,my(v); if(mapisdefined(memoA346241,n,&v), v, v = -sumdiv(n,d,if(d<n,A003415(n/d)*A346241(d),0)); mapput(memoA346241,n,v); (v)));
%Y A346241 Cf. A003415, A354806, A354807, A354808 (positions of negative terms), A354809 (of terms >= 0), A354818 (of even terms).
%Y A346241 Cf. also A317835, A347082, A347084, A354821.
%K A346241 sign
%O A346241 1,4
%A A346241 _Antti Karttunen_, Jul 13 2021