A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

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

Offset

1,16

Comments

First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.

The number of squares dividing the largest exponentially odd divisor of n is A325837(n).

The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]

The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - Amiram Eldar, Sep 08 2023

Links

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

Formula

a(n) = A325837(A008833(n)).

a(n) = 1 if and only if n is a biquadratefree number (A046100).

Multiplicative with a(p^e) = max(1, floor(e/2)).

Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x -> max(1, x\2), factor(n)[, 2]));

Crossrefs

Cf. A008833, A046100, A046951, A325837, A365334.

Cf. A043289, A053164, A063775, A203640, A295658.

Sequence in context: A043289 A063775 A053164 * A295658 A307427 A318672

Adjacent sequences: A365330 A365331 A365332 * A365334 A365335 A365336

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 01 2023

Extensions

Name corrected by Amiram Eldar, Sep 08 2023

Status

approved