A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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

Offset: 1

Author

Amiram Eldar, Jun 16 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A367515 at n = 128.

The sum of these divisors is A385043(n), and the largest of them is A367168(n).

Links

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

Crossrefs

The unitary analog of A353898.

Cf. A005117, A034444, A138302, A209229, A367168, A367515, A385043.

The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), this sequence (exponentially 2^n), A385044 (5-rough).

Programs

Mathematica

f[p_, e_] := Boole[e == 2^IntegerExponent[e, 2]] + 1; a[ 1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> (x == 1<
				

Formula

Multiplicative with a(p^e) = A209229(e) + 1.

a(n) <= A034444(n), with equality if and only if n is in A138302.

a(n) <= A353898(n), with equality if and only if n is squarefree (A005117).