A383123 The Möbius transform of A382883.

1, -2, -2, 2, -2, 4, -2, 0, 2, 4, -2, -3, -2, 4, 4, -1, -2, -3, -2, -3, 4, 4, -2, 0, 2, 4, 0, -3, -2, -8, -2, 1, 4, 4, 4, 0, -2, 4, 4, 0, -2, -8, -2, -3, -3, 4, -2, 1, 2, -3, 4, -3, -2, 0, 4, 0, 4, 4, -2, 5, -2, 4, -3, -2, 4, -8, -2, -3, 4, -8, -2, 1, -2, 4

Offset

1,2

Comments

See the comments in A382883.

Links

Table of n, a(n) for n=1..74.

Formula

a(n) = Sum_{d|n} A382883(n/d)*Moebius(d).

Maple

a := proc(n) with(NumberTheory): local d;
add(A382883(iquo(n, d))*Moebius(d), d in Divisors(n)) end:
seq(a(n), n = 1..74);

Mathematica

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; a[n_] := DivisorSum[n, MoebiusMu[#] * f[n/#] &]; Array[a, 100] (* Amiram Eldar, Apr 29 2025 *)

Prog

(SageMath)  # Implemented as the value of a transform.
from typing import Callable
def A382883Trans(b: Callable[[int], int]) -> Callable[[int], int]:
    @cached_function
    def vb(n: int) -> int:
        return sum(A382883(n//d)*b(d) for d in divisors(n))
    return vb
A383123 = A382883Trans(moebius)
print([A383123(n) for n in range(1, 75)])

Crossrefs

Cf. A382883, A008683.

Sequence in context: A132003 A122857 A109810 * A365498 A380089 A367515

Adjacent sequences: A383120 A383121 A383122 * A383124 A383125 A383126

Keyword

sign

Author

Peter Luschny, Apr 18 2025

Status

approved