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

1,4

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

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).

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]

Prog

(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))); }

Crossrefs

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

Cf. A001620, A002117, A244115.

Sequence in context: A370783 A295295 A365336 * A069290 A365334 A348660

Adjacent sequences: A365401 A365402 A365403 * A365405 A365406 A365407

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 03 2023

Status

approved