A386470 The number of divisors of n whose exponents in their prime factorization are squares.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

Offset

1,2

Comments

First differs from A365171 and A369310 at n = 32.

First differs from A365488 at n = 128.

The number of terms in A197680 that divide n.

The sum of these divisors is A386471(n) and the largest of them is A386469(n).

Links

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

Formula

Multiplicative with a(p^e) = A048760(e) + 1.

a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A005117, A197680, A365171, A365488, A369310, A386469, A386471.

Sequence in context: A365491 A365210 A365488 * A365171 A369310 A318465

Adjacent sequences: A386467 A386468 A386469 * A386471 A386472 A386473

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jul 22 2025

Status

approved