A365332 The sum of divisors of the largest square dividing n.

1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 7, 31, 1, 13, 7, 1, 1, 1, 31, 1, 1, 1, 91, 1, 1, 1, 7, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 13, 1, 7, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 31, 7, 1, 1, 1, 31, 121

Offset

1,4

Comments

All the terms are odd.

The number of these divisors is A365331(n).

The sum of divisors of the square root of the largest square dividing n is A069290(n).

Links

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

Formula

a(n) = A000203(A008833(n)).

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

Multiplicative with a(p^e) = (p^(e + 1 - (e mod 2)) - 1)/(p - 1).

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

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 5*zeta(3/2)/Pi^2 = 1.323444812234... .

Mathematica

f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - 1)/(p - 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~, (f[i, 1]^(f[i, 2] + 1 - f[i, 2]%2) - 1)/(f[i, 1] - 1)); }

Crossrefs

Cf. A000203, A005117, A008833, A069290, A078434, A365331.

Sequence in context: A344696 A336457 A271498 * A367483 A348281 A317940

Adjacent sequences: A365329 A365330 A365331 * A365333 A365334 A365335

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 01 2023

Status

approved