A391988 The number of exponential divisors of the exponentially squarefree numbers.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2

Offset

1,4

Comments

First differs from A388973 at n = 246.

Also, the number of exponential unitary divisors of the exponentially squarefree numbers.

Links

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

Formula

a(n) = A049419(A209061(n)).

a(n) = A278908(A209061(n)).

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=2} (A074823(k) - A074823(k-1))/p^k) = 1.49848918426328501463..., where d = A262276.

Mathematica

s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Boole[AllTrue[e, SquareFreeQ]] * Times @@ (2^PrimeNu[e])]; s[1] = 1; Select[Array[s, 100], # > 0 &]

Prog

(PARI) f(n) = {my(e = factor(n)[, 2]); for(i = 1, #e, if(!issquarefree(e[i]), return(0))); vecprod(apply(x -> 2^omega(x), e)); }
list(lim) = select(x -> x > 0, vector(lim, i, f(i)));

Crossrefs

Cf. A049419, A074823, A209061, A262276, A278908, A388973, A391989.

Sequence in context: A392450 A394425 A388973 * A007424 A369163 A380398

Adjacent sequences: A391985 A391986 A391987 * A391989 A391990 A391991

Keyword

nonn,easy

Author

Amiram Eldar, Dec 26 2025

Status

approved