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

1, -1, -1, -3, -1, -3, -1, -5, -5, -5, -1, -1, -1, -7, -6, -3, -1, -2, -1, -5, -8, -11, -1, 17, -9, -13, -16, -9, -1, 3, -1, 11, -12, -17, -10, 33, -1, -19, -14, 19, -1, 1, -1, -17, -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

Offset: 1

Author

Antti Karttunen, Jul 13 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A003415, A354806, A354807, A354808 (positions of negative terms), A354809 (of terms >= 0), A354818 (of even terms).

Cf. also A317835, A347082, A347084, A354821.

Programs

PARI

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415plusA063524(n) = if(n<=1, 1, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v346241 = DirInverseCorrect(vector(up_to,n,A003415plusA063524(n)));
A346241(n) = v346241[n];

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
memoA346241 = Map();
A346241(n) = if(1==n,1,my(v); if(mapisdefined(memoA346241,n,&v), v, v = -sumdiv(n,d,if(dA003415(n/d)*A346241(d),0)); mapput(memoA346241,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA003415(n/d) * a(d).