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URL: https://oeis.org/A389775

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A389775
Absolute value of the inverse Möbius transform of mu(n)*phi(n^2).
5
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
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
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.
MATHEMATICA
f[p_, e_] := p^2 - p - 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Oct 15 2025 *)
PROG
(PARI) a(n)=(-1)^omega(n)*sumdiv(n, d, moebius(d)*eulerphi(d^2));
CROSSREFS
Cf. A000010 (phi), A001221 (omega), A008683 (mu), A008966 (mu^2), A202535.
Related arithmetic functions: A367865, A367866, A389977, A389978.
Sequence in context: A151335 A297174 A226605 * A055584 A193861 A349544
KEYWORD
nonn,mult,easy
AUTHOR
Aloe Poliszuk, Oct 14 2025
STATUS
approved