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

1,4

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

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... .

Mathematica

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

Prog

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

Crossrefs

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

Cf. A081117.

Sequence in context: A364209 A268238 A081117 * A129252 A327936 A333748

Adjacent sequences: A368331 A368332 A368333 * A368335 A368336 A368337

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved