Correlating the von Mangoldt function with periodic sequences, or the expansion of itself really

The Dirichlet inverse of the Euler totient function is:
$$varphi^{-1}(n) = sum_{d mid n} mu(d)d tag{1}$$
Consider the sequences:
$$a(n)=sum _{k=1}^{j} frac{varphi^{-1}(gcd (n,k))}{k}$$
$$b(n)=sum _{k=1}^{j+1} frac{varphi^{-1}(gcd (n,k))}{k}$$

Question:

For what values of $j$ does $b(n)$ eventually correlate better than $a(n)$ with the von Mangoldt function $Lambda(n)$ as $n rightarrow infty$?

The exceptions of $j$ when $b(n)$ correlates worse than $a(n)$ appears to be:
$$j=7, 15, 24, 26, 31$$ which when added with $1$ gives initially a sequence of powers of some sort:
$$j+1=8, 16, 25, 27, 32$$

A very slow Mathematica program that computes the sequences and compares their Pearson correlation with the von Mangoldt function is:

Clear(a, n, k, start, end)
nnn = 400;
a(n_) := Total(Divisors(n)*MoebiusMu(Divisors(n)));
Do(
 column = j;
 earlier = 
  Table(Correlation(
    Table(Sum(a(GCD(n, k))/k, {k, 1, column}), {n, 1, nn}), 
    Table(N(MangoldtLambda(n)), {n, 1, nn})), {nn, 2, nnn});
 later = Table(
   Correlation(
    Table(Sum(a(GCD(n, k))/k, {k, 1, column + 1}), {n, 1, nn}), 
    Table(N(MangoldtLambda(n)), {n, 1, nn})), {nn, 2, nnn});
 sign = Sign(later - earlier);
 Print(Count(sign, 1)), {j, 2, 32})

nnn = 400; is considered a large number that serves as the substitute for infinity in the program.