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

Author

Amiram Eldar, Sep 01 2023

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

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

Programs

Mathematica

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

PARI

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

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... .

Extensions

Name corrected by Amiram Eldar, Sep 08 2023