A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

First differs from A278908, A307848, A323308 and A358260 at n = 64.

The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n).

Links

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

Formula

a(n) = A322483(A000188(n)).

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

Multiplicative with a(p^e) = 2 + floor((e-2)/4).

Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .

Mathematica

f[p_, e_] := 2 + Floor[(e-2)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> 2 + (x-2)\4, factor(n)[, 2]));

Crossrefs

Cf. A000188, A005117, A013662, A046951, A322483.

Cf. A278908, A307848, A323308, A358260.

Sequence in context: A007424 A369163 A323308 * A278908 A307848 A358260

Adjacent sequences: A365546 A365547 A365548 * A365550 A365551 A365552

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 08 2023

Status

approved