A359577 Dirichlet inverse of A085731, where A085731 is the greatest common divisor of n and the arithmetic derivative of n.

1, -1, -1, -3, -1, 1, -1, 3, -2, 1, -1, 3, -1, 1, 1, -3, -1, 2, -1, 3, 1, 1, -1, -3, -4, 1, -22, 3, -1, -1, -1, 3, 1, 1, 1, 6, -1, 1, 1, -3, -1, -1, -1, 3, 2, 1, -1, 3, -6, 4, 1, 3, -1, 22, 1, -3, 1, 1, -1, -3, -1, 1, 2, -3, 1, -1, -1, 3, 1, -1, -1, -6, -1, 1, 4, 3, 1, -1, -1, 3, 28, 1, -1, -3, 1, 1, 1, -3, -1, -2, 1, 3, 1, 1, 1, -3, -1, 6, 2, 12, -1, -1, -1, -3, -1

Offset: 1

Author

Antti Karttunen, Jan 06 2023

Keywords

• sign
• mult

Comments

Multiplicative because A085731 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A085731, A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).

Programs

Maple

g:= proc(n) option remember;
      igcd(n, n*add(i[2]/i[1], i=ifactors(n)[2]))
    end:
a:= proc(n) option remember; `if`(n=1, 1, -add(
      a(d)*g(n/d), d=numtheory[divisors](n) minus {n}))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 07 2023

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); s[n_] := GCD[n, d[n]]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jan 07 2023 *)

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA085731(n/d) * a(d).