A389775 - OEIS

A389775

Absolute value of the inverse Möbius transform of mu(n)*phi(n^2).

1, 1, 5, 1, 19, 5, 41, 1, 5, 19, 109, 5, 155, 41, 95, 1, 271, 5, 341, 19, 205, 109, 505, 5, 19, 155, 5, 41, 811, 95, 929, 1, 545, 271, 779, 5, 1331, 341, 775, 19, 1639, 205, 1805, 109, 95, 505, 2161, 5, 41, 19, 1355, 155, 2755, 5, 2071, 41, 1705, 811, 3421, 95

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OFFSET

1,3

COMMENTS

From Aloe Poliszuk, Nov 29 2025: (Start)

Also the absolute value of the inverse Möbius transform of n * mu(n) * phi(n).

One of four arithmetic functions multiplicative by a monic degree 2 Littlewood polynomial of p. The other three are A367865, A367866, and A389977. (End)

LINKS

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

Wikipedia, Littlewood polynomial.

FORMULA

Multiplicative with a(p^e) = p^2 - p - 1.

a(n) = a(rad(n)), where rad = A007947.

a(n) = (-1)^A001221(n) * Sum_{d|n} d * mu(d)^2 * Sum_{k|d} k * mu(k).

a(n) = (-1)^A001221(n) * Sum_{d|n} mu(d) * phi(d^2).

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 2/p + (2*p)/(1+p+p^2)) = 0.3782601715591552459004... . - Amiram Eldar, Oct 15 2025

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(2-s) - p^(1-s) - 2*p^(-s)). - Aloe Poliszuk, Nov 29 2025

EXAMPLE

a(24) = a(8)*a(3) = (4 - 2 - 1)*(9 - 3 - 1) = 5.