A365404 The sum of the unitary divisors of the square root of the largest square dividing n.

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 5, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1

Offset: 1

Author

Amiram Eldar, Sep 03 2023

Keywords

• nonn
• easy
• mult

Comments

The number of these divisors is A323308(n).

The sum of the unitary divisors of the largest square dividing n is A365403(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A000188, A034448, A323308, A365403.

Cf. A001620, A002117, A244115.

Programs

Mathematica

f[p_, e_] := If[e == 1, 1, p^Floor[e/2] + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + f[i,1]^(f[i,2]\2)));}

Formula

a(n) = A034448(A000188(n)).

a(n) >= 1 with equality if and only if n is squarefree (A005117).

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

Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-1).

Sum_{k=1..n} a(k) ~ (n/(2*zeta(3))) * (log(n) + 3*gamma - 1 - 4*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).