A365335 The number of exponentially odd coreful divisors of the square root of the largest square dividing n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,64

Comments

First differs from A160338 at n = 64, and from A178489 at n = 65.

The number of divisors of the square root of the largest square dividing n is A046951(n).

The number of exponentially odd divisors of the square root of the largest square dividing n is A365549(n) and their sum is A365336(n). [corrected, Sep 08 2023]

Links

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

Formula

a(n) = A325837(A000188(n)).

a(n) > 1 if and only if n is a bicubeful number (A355265).

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

Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 - 1/p^(4*s) + 1/p^(6*s)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.0181534831085... .

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x -> max(1, (x+2)\4), factor(n)[, 2]));

Crossrefs

Cf. A000188, A046951, A325837, A355265, A365336, A365549.

Cf. A160338, A178489.

Sequence in context: A025452 A299202 A194337 * A376663 A299912 A369162

Adjacent sequences: A365332 A365333 A365334 * A365336 A365337 A365338

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 01 2023

Extensions

Name corrected by Amiram Eldar, Sep 08 2023

Status

approved