A359793 Dirichlet inverse of (-1)^A003415(n), where A003415 is the arithmetic derivative of n.

1, 1, 1, 0, 1, 3, 1, -2, 0, 3, 1, 2, 1, 3, 1, -4, 1, 4, 1, 2, 1, 3, 1, -6, 0, 3, 0, 2, 1, 9, 1, -4, 1, 3, 1, 8, 1, 3, 1, -6, 1, 9, 1, 2, 0, 3, 1, -20, 0, 4, 1, 2, 1, 4, 1, -6, 1, 3, 1, 16, 1, 3, 0, 0, 1, 9, 1, 2, 1, 9, 1, -4, 1, 3, 0, 2, 1, 9, 1, -20, 0, 3, 1, 16, 1, 3, 1, -6, 1, 12, 1, 2, 1, 3, 1, -28, 1, 4, 0

Offset: 1

Author

Antti Karttunen, Jan 14 2023

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A359792.

Cf. A008966, A005117, A013929 (apparently parity of terms, positions of odd terms, and positions of even terms).

Cf. also A359780, A359823.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A359792(n) = ((-1)^A003415(n));
memoA359793 = Map();
A359793(n) = if(1==n,1,my(v); if(mapisdefined(memoA359793,n,&v), v, v = -sumdiv(n,d,if(dA359792(n/d)*A359793(d),0)); mapput(memoA359793,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA359792(n/d) * a(d).