A369974 Dirichlet inverse of A369001, where A369001(n) = 1 if n' / gcd(n,n') is even, otherwise 0, and n' stands for the arithmetic derivative of n, A003415.

1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, -1, -1, 0, 0, 0, -1, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, -1, -1, 0, -1, 0

Offset: 1

Author

Antti Karttunen, Feb 09 2024

Keywords

• sign

Comments

a(144) = 2 is the first term > 1.

Links

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

Crossrefs

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

Agrees paritywise with A369978.

Cf. A358777, A359763, A359773, A359780 for similar sequences.

Programs

PARI

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369001(n) = !(A083345(n)%2);
memoA369974 = Map();
A369974(n) = if(1==n,1,my(v); if(mapisdefined(memoA369974,n,&v), v, v = -sumdiv(n,d,if(dA369001(n/d)*A369974(d),0)); mapput(memoA369974,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA369001(n/d) * a(d).