A365800 Dirichlet inverse of the characteristic function of triangular numbers (A010054).

1, 0, -1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -3, -1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1

Offset: 1

Author

Antti Karttunen, Oct 05 2023

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..101025

Crossrefs

Cf. A000217, A010054, A365799 (parity of terms).

Absolute values differ from A308264.

Programs

PARI

A010054(n) = issquare(8*n + 1);
memoA365800 = Map();
A365800(n) = if(1==n,1,my(v); if(mapisdefined(memoA365800,n,&v), v, v = -sumdiv(n,d,if(dA010054(n/d)*A365800(d),0)); mapput(memoA365800,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA010054(n/d) * a(d).