A368885 The number of unitary divisors of n that are squares of a squarefree number (A062503).

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

Offset

1,4

Comments

First differs from A294932 at n = 32.

The largest of these divisors is A368884(n).

Links

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

Formula

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

a(n) >= 1, with equality if and only if n is in A337050.

a(n) <= A034444(n), with equality if and only if n is in A062503.

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(3*s)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.30596827416754083231... .

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x->if(x==2, 2, 1), factor(n)[, 2]));
(Python)
from sympy import factorint
def A368885(n): return 1<<sum(1 for e in factorint(n).values() if e==2) # Chai Wah Wu, Jan 09 2024

Crossrefs

Cf. A034444, A062503, A294932, A337050, A368884.

Sequence in context: A277231 A122934 A072170 * A294932 A327804 A056624

Adjacent sequences: A368882 A368883 A368884 * A368886 A368887 A368888

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 09 2024

Status

approved