A353468 Dirichlet inverse of A353467.

1, 1, -1, -1, 1, 1, -1, 2, 2, -1, 1, -3, -1, 1, -3, -5, 1, -1, -1, 3, 3, -1, 1, 10, -1, 1, -5, -3, -1, 1, 1, 14, -3, -1, 1, 0, -1, 1, 3, -10, 1, -1, -1, 3, 10, -1, 1, -35, 2, 2, -3, -3, -1, 2, -1, 10, 3, 1, 1, 3, -1, -1, -10, -42, 1, 1, 1, 3, -3, -3, -1, 10, 1, 1, 0, -3, -3, -1, -1, 35, 14, -1, 1, -3, -1, 1, 3, -10

Offset: 1

Author

Antti Karttunen, Apr 21 2022

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000040, A252463, A353467 [Dirichlet inverse], A353469 [sum with it].

Cf. also A353458.

Programs

PARI

A252463(n) = if(!(n%2),n/2,my(f=factor(n)); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f));
memoA353467 = Map();
A353467(n) = if(1==n,1,my(v); if(mapisdefined(memoA353467,n,&v), v, v = -sumdiv(n,d,if(dA353467(A252463(n/d))*A353467(d),0)); mapput(memoA353467,n,v); (v)));
A353468(n) = A353467(A252463(n));

Formula

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA353467(n/d) * a(d).

a(n) = A353467(A252463(n)).

For all n >= 1, a(A000040(n)) = ((-1)^(n-1)).