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

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 17, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset

1,4

Comments

The number of these divisors is A323308(n).

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

Links

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

Formula

a(n) = A034448(A008833(n)).

a(n) <= A034448(n) with equality if and only if n is a square (A000290).

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

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

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

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(4)) = 30*zeta(3/2)/Pi^4 = 0.804557969165... .

Mathematica

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

Prog

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

Crossrefs

Cf. A000290, A005117, A008833, A034448, A323308, A365404.

Cf. A013662, A078434.

Sequence in context: A351086 A102280 A370239 * A373439 A035316 A385130

Adjacent sequences: A365400 A365401 A365402 * A365404 A365405 A365406

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 03 2023

Status

approved