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

1,8

Comments

First differ from A043281 at n = 49.

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(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... .

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

Cf. A043281.

Sequence in context: A375667 A341644 A198293 * A043281 A365634 A320267

Adjacent sequences: A368327 A368328 A368329 * A368331 A368332 A368333

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 21 2023

Status

approved