A081117 -id:A081117 - cp's OEIS frontend

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

Amiram Eldar, Dec 21 2023

First differs from A081117 at n = 28.

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

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

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

Cf. A081117.

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

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

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... .

A081129 Differences of Beatty sequence for cube root of 3.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Apr 16 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A059539, A081117, A081147, A081168.

A081129:= func< n | Floor((n+1)*3^(1/3)) - Floor(n*3^(1/3)) >;
[A081129(n): n in [0..120]]; // G. C. Greubel, Jan 15 2024

Differences[Floor[Range[0,110]Surd[3,3]]] (* Harvey P. Dale, Apr 06 2022 *)

a(n)=floor((n+1)*3^(1/3))-floor(n*3^(1/3))

def A081129(n): return floor((n+1)*3^(1/3)) - floor(n*3^(1/3))
[A081129(n) for n in range(121)] # G. C. Greubel, Jan 15 2024

a(n) = floor((n+1)*3^(1/3)) - floor(n*3^(1/3)).

cp's OEIS Frontend

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

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