A383104 Inverse Möbius transform of A382883.

1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0

Offset: 1

Author

Peter Luschny, Apr 16 2025

Keywords

• nonn

Crossrefs

Cf. A382883, A366988.

Programs

Maple

# Seen as a special case of a transformation:
A382883Transform := (b, len) -> local n, d; seq(add(A382883(n/d)*b(d), d in numtheory:-divisors(n)), n = 1..len): A382883Transform(n -> 1, 99);

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, f[#] &]; Array[a, 100] (* Amiram Eldar, Apr 29 2025 *)

SageMath

def a(n): return sum(A382883(n/d) for d in divisors(n))
print([a(n) for n in range(1, 90)])
# More general:
def A382883Transform(n: int, b: Callable[[int], int]) -> int:
    return sum(A382883(n/d)*b(d) for d in divisors(n))
def a(n) -> int: return A382883Transform(n, lambda x: 1)

Formula

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