A347084 - OEIS

%I #11 Nov 15 2021 01:32:00

%S 1,-3,-4,1,-6,13,-8,1,1,19,-12,-6,-14,25,25,1,-18,-5,-20,-8,33,37,-24,

%T -5,1,43,2,-10,-30,-87,-32,1,49,55,49,6,-38,61,57,-7,-42,-113,-44,-14,

%U -8,73,-48,-4,1,-5,73,-16,-54,-9,73,-9,81,91,-60,51,-62,97,-10,1,85,-165,-68,-20,97,-163,-72,2,-74,115

%N Dirichlet inverse of A129283, n + A003415(n).

%H Antti Karttunen, <a href="/A347084/b347084.txt">Table of n, a(n) for n = 1..16384</a>

%F a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, d<n} a(d) * A129283(n/d).

%F a(n) = A347085(n) - A129283(n).

%F a(n) = A347082(n) - A347086(n).

%o (PARI)

%o up_to = 16384;

%o 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 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));

%o A347084(n) = v347084[n];

%Y Cf. A003415, A129283, A347082, A347085, A347086, A348995 (positions of 1's).

%Y Cf. also A346241, A348976.

%K sign

%O 1,2

%A _Antti Karttunen_, Aug 17 2021