A368330 The number of terms of A054743 that are unitary divisors of n.

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

Offset: 1

Author

Amiram Eldar, Dec 21 2023

Keywords

• nonn
• easy
• mult

Comments

First differ from A043281 at n = 49.

Links

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

Crossrefs

Cf. A034444, A054743, A207481, A368328, A368329, A368331, A368334.

Cf. A043281.

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(A368329(n)).

a(n) >= 1, with equality if and only if n is in A207481.

a(n) <= A034444(n), with equality if and only if n is in A054743.

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^(p+1)) = 1.13896197534988330925... .