A383762 The number of unitary divisors of n that are exponentially squarefree numbers.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset

1,2

Comments

First differs from A365499 at n = 32.

The sum of these divisors is A383763(n) and the largest of them is A383764(n).

Links

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

Formula

Multiplicative with a(p^e) = 2 if e is squarefree (A005117), and 1 otherwise.

a(n) <= A034444(n), with equality if and only if n is an exponentially squarefree number (A209061).

a(n) <= A365680(n), with equality if and only if n is a squarefree number.

Mathematica

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

Prog

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

Crossrefs

Cf. A005117, A034444, A077610, A209061, A365499, A365680, A383763, A383764.

Sequence in context: A389781 A048669 A365499 * A158522 A034444 A365491

Adjacent sequences: A383759 A383760 A383761 * A383763 A383764 A383765

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 09 2025

Status

approved