A344465 a(n) = Sum_{d|n} d^(mu(d)^2).

1, 3, 4, 4, 6, 12, 8, 5, 5, 18, 12, 14, 14, 24, 24, 6, 18, 14, 20, 20, 32, 36, 24, 16, 7, 42, 6, 26, 30, 72, 32, 7, 48, 54, 48, 17, 38, 60, 56, 22, 42, 96, 44, 38, 26, 72, 48, 18, 9, 20, 72, 44, 54, 16, 72, 28, 80, 90, 60, 76, 62, 96, 34, 8, 84, 144, 68, 56, 96, 144, 72

Offset

1,2

Comments

If p is prime, a(p) = Sum_{d|p} d^(mu(d)^2) = 1^1 + p^1 = p + 1.

Inverse Möbius transform of n^(mu(n)^2) (A385212). - Wesley Ivan Hurt, Mar 31 2025

Links

Table of n, a(n) for n=1..71.

Formula

a(n) = sigma(rad(n)) + tau(n) - 2^omega(n). - Wesley Ivan Hurt, Jun 20 2025

a(p^k) = p + k for p prime and k >= 1. - Wesley Ivan Hurt, Jul 02 2025

Example

a(10) = Sum_{d|10} d^(mu(d)^2) = 1^1 + 2^1 + 5^1 + 10^1 = 18.

Mathematica

Table[Sum[k^Abs[MoebiusMu[k]] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

Crossrefs

Cf. A000005 (tau), A008683 (mu), A034444, A048250, A385212.

Sequence in context: A086659 A265887 A345264 * A008473 A326043 A069088

Adjacent sequences: A344462 A344463 A344464 * A344466 A344467 A344468

Keyword

nonn

Author

Wesley Ivan Hurt, May 20 2021

Status

approved