# 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.

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