A369978 Dirichlet inverse of sequence b(n) = 1+A083345(n), where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

1, -2, -2, 2, -2, 2, -2, -4, 1, 0, -2, 3, -2, -2, -1, 9, -2, 4, -2, 9, -3, -6, -2, -8, 1, -8, 2, 15, -2, 12, -2, -18, -7, -12, -5, -14, -2, -14, -9, -22, -2, 18, -2, 27, 10, -18, -2, 20, 1, 10, -13, 33, -2, -8, -9, -36, -15, -24, -2, -16, -2, -26, 14, 36, -11, 30, -2, 45, -19, 16, -2, 22, -2, -32, 12, 51, -11, 36

Offset: 1

Author

Antti Karttunen, Feb 09 2024

Keywords

• sign

Links

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

Crossrefs

Cf. A003415, A083345, A369001, A369974, A369975 (parity of terms), A369976 (positions of odd terms).

Cf. A359790 and A366265 for similar sequences.

Programs

PARI

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
memoA369978 = Map();
A369978(n) = if(1==n,1,my(v); if(mapisdefined(memoA369978,n,&v), v, v = -sumdiv(n,d,if(dA083345(n/d))*A369978(d),0)); mapput(memoA369978,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083345(n/d)) * a(d).