A354348 Dirichlet inverse of function f(1) = 1, f(n) = gcd(A003415(n), A276086(n)) for n > 1.

1, -1, -1, 0, -1, -3, -1, -2, -5, 1, -1, 8, -1, -1, 0, 4, -1, 18, -1, -2, -8, 1, -1, 0, -9, -13, 8, 4, -1, 9, -1, -2, -12, 1, -4, -9, -1, -19, 0, 8, -1, 29, -1, -2, 10, -23, -1, 2, -13, 4, -8, 22, -1, 20, 0, 2, 0, 1, -1, -13, -1, -1, 26, 2, -16, 33, -1, -2, 0, 13, -1, -14, -1, -1, 16, 36, -16, 37, -1, -10, 19, 1

Offset: 1

Author

Antti Karttunen, Jun 08 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030

Crossrefs

Cf. A003415, A276086, A327858.

Cf. also A346242, A354347.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n), A276086(n));
memoA354348 = Map();
A354348(n) = if(1==n,1,my(v); if(mapisdefined(memoA354348,n,&v), v, v = -sumdiv(n,d,if(dA327858(n/d)*A354348(d),0)); mapput(memoA354348,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA327858(n/d) * a(d).