A347085 -id:A347085 - cp's OEIS frontend

A347084 Dirichlet inverse of A129283, n + A003415(n).

1, -3, -4, 1, -6, 13, -8, 1, 1, 19, -12, -6, -14, 25, 25, 1, -18, -5, -20, -8, 33, 37, -24, -5, 1, 43, 2, -10, -30, -87, -32, 1, 49, 55, 49, 6, -38, 61, 57, -7, -42, -113, -44, -14, -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

Offset: 1

Antti Karttunen, Aug 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Cf. also A346241, A348976.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
A347084(n) = v347084[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA129283(n/d).
a(n) = A347085(n) - A129283(n).
a(n) = A347082(n) - A347086(n).

A347083 Sum of -A168036 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, -1, 4, 8, 0, -4, 0, 12, 16, -7, 0, -2, 0, -6, 24, 20, 0, -19, 16, 24, 4, -8, 0, -14, 0, -21, 40, 32, 48, -18, 0, 36, 48, -33, 0, -18, 0, -12, 4, 44, 0, -58, 36, 6, 64, -14, 0, -22, 80, -47, 72, 56, 0, -29, 0, 60, 8, -47, 96, -26, 0, -18, 88, -22, 0, -62, 0, 72, 20, -20, 120, -30, 0, -108, -11, 80

Offset: 1

Antti Karttunen, Aug 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A168036, A347082, A347085.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A168036(n) = (A003415(n)-n);
v347082 = DirInverseCorrect(vector(up_to,n,-A168036(n)));
A347082(n) = v347082[n];
A347083(n) = (A347082(n)-A168036(n));

a(n) = A347082(n) - A168036(n).

For n > 1, a(n) = Sum_{d|n, 1A168036(d) * A347082(n/d).

cp's OEIS Frontend

A347084 Dirichlet inverse of A129283, n + A003415(n).

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