A368334 The number of terms of A054744 that are unitary divisors of n.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 21 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A081117 at n = 28.

Also, the number of terms of A072873 that are unitary divisors of n.

Links

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

Crossrefs

Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.

Cf. A081117.

Programs

Mathematica

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

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, 2));}

Formula

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

a(n) = A034444(A368333(n)).

a(n) = A034444(A327939(n)).

a(n) >= 1, with equality if and only if n is in A048103.

a(n) <= A034444(n), with equality if and only if n is in A054744.

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .