A363289 Sum of the divisor complements of the unitary squarefree divisors of n.

1, 3, 4, 4, 6, 12, 8, 8, 9, 18, 12, 16, 14, 24, 24, 16, 18, 27, 20, 24, 32, 36, 24, 32, 25, 42, 27, 32, 30, 72, 32, 32, 48, 54, 48, 36, 38, 60, 56, 48, 42, 96, 44, 48, 54, 72, 48, 64, 49, 75, 72, 56, 54, 81, 72, 64, 80, 90, 60, 96, 62, 96, 72, 64, 84, 144, 68, 72, 96, 144, 72

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = n * Sum_{d|n, gcd(d,n/d)=1} mu(d)^2 / d.

From Amiram Eldar, Oct 31 2025: (Start)

Multiplicative with a(p) = p + 1, and a(p^e) = p^e for e >= 2.

Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s-1)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.30596827416754083231... . (End)

Maple

f:= proc(n) local S, d;
  S:= select(t -> igcd(t, n/t) = 1 and numtheory:-issqrfree(t), numtheory:-divisors(n));
  n * add(1/d, d = S)
end proc:
map(f, [$1..100]); # Robert Israel, Oct 29 2025

Mathematica

a[n_] := DivisorSum[n, n/# &, CoprimeQ[#, n/#] && SquareFreeQ[#] &]; Array[a, 100] (* Amiram Eldar, May 26 2023 *)
f[p_, e_] := p^e + If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2025 *)

Prog

(PARI) a(n) = sumdiv(n, d, if ((gcd(d, n/d) == 1) && issquarefree(d), n/d)); \\ Michel Marcus, May 26 2023

Crossrefs

Cf. A092261.

Sequence in context: A178450 A378435 A325973 * A019462 A078071 A359789

Adjacent sequences: A363286 A363287 A363288 * A363290 A363291 A363292

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, May 26 2023

Status

approved