A383210 The Dirichlet inverse of A382883.

1, 1, 1, 0, 1, 1, 1, -2, 0, 1, 1, -1, 1, 1, 1, -3, 1, -1, 1, -1, 1, 1, 1, -5, 0, 1, -2, -1, 1, 1, 1, -2, 1, 1, 1, -2, 1, 1, 1, -5, 1, 1, 1, -1, -1, 1, 1, -6, 0, -1, 1, -1, 1, -5, 1, -5, 1, 1, 1, -3, 1, 1, -1, 3, 1, 1, 1, -1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, 1

Offset

1,8

Comments

See the comments in A382883.

Links

Table of n, a(n) for n=1..79.

Formula

a(n) = -Sum_{d|n, d<n} a(d)*A382883(n/d) for n >= 2, otherwise 1.

Maple

with(NumberTheory): a := proc(n) option remember; local d; ifelse(n < 2, n, -add(a(d)*A382883(iquo(n, d)), d in Divisors(n) minus {n})) end: seq(a(n), n = 1..79);

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_] := a[n] = -DivisorSum[n, a[#] * f[n/#] &, # < n &]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Apr 29 2025 *)

Prog

(SageMath)
from typing import Callable
def iT(b: Callable[[int], int]) -> Callable[[int], int]:
    @cached_function
    def g(n: int) -> int:
        if n == 1:
            return 1
        s = sum(g(n//d)*b(d) for d in divisors(n)[1:])
        return -s
    return g
A383210 = iT(A382883)
print([A383210(n) for n in range(1, 80)])

Crossrefs

Cf. A382883.

Sequence in context: A203949 A070200 A359833 * A025914 A376631 A284977

Adjacent sequences: A383207 A383208 A383209 * A383211 A383212 A383213

Keyword

sign

Author

Peter Luschny, Apr 19 2025

Status

approved