A389977 Absolute value of the inverse Möbius transform of n*mu(n)*sigma(n).

1, 5, 11, 5, 29, 55, 55, 5, 11, 145, 131, 55, 181, 275, 319, 5, 305, 55, 379, 145, 605, 655, 551, 55, 29, 905, 11, 275, 869, 1595, 991, 5, 1441, 1525, 1595, 55, 1405, 1895, 1991, 145, 1721, 3025, 1891, 655, 319, 2755, 2255, 55, 55, 145, 3355, 905, 2861, 55, 3799, 275, 4169, 4345, 3539, 1595

Offset

1,2

Comments

Also the absolute value of the inverse Möbius transform of mu(n) * psi(n^2), where psi = A001615. - Aloe Poliszuk, Nov 29 2025

One of four arithmetic functions multiplicative by a monic degree 2 Littlewood polynomial of p. The other three are A367865, A367866, and A389775.

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)).

a(n) = (-1)^A001221(n) * Sum_{d|n} d * mu(d) * A048250(d).

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

Sum_{k=1..n} a(k) ~ c * zeta(3) * n^3 / 3, where c = Product_{p prime} (1 - 3/p^3 + 2/p^4) = 0.662190331763714968705... . - Amiram Eldar, Oct 21 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(6) = a(2)*a(3) = (4 + 2 - 1)*(9 + 3 - 1) = 55.

Mathematica

f[p_, e_] := p^2 + p - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Oct 21 2025 *)

Prog

(PARI) a(n) = (-1)^omega(n)*sumdiv(n, d, d*moebius(d)*sigma(d));

Crossrefs

Cf. A000203 (sigma), A001221 (omega), A001615 (psi), A007947 (rad), A008683 (mu), A008966 (mu^2).

Cf. A023900, A048250.

Related arithmetic functions: A202535, A367865, A367866, A389775, A389978.

Sequence in context: A110353 A377050 A097720 * A077806 A365731 A019305

Adjacent sequences: A389974 A389975 A389976 * A389978 A389979 A389980

Keyword

nonn,mult,easy

Author

Aloe Poliszuk, Oct 20 2025

Status

approved