A392450 The number of exponential unitary (or e-unitary) divisors of n that are exponentially squarefree numbers (A209061).

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

Offset

1,4

Comments

First differs from A366309 at n = 32.

Exponential unitary divisors are defined in A278908.

Links

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

Formula

Multiplicative with a(p^e) = A056671(e) (the number of squarefree unitary divisors of e).

a(n) <= A278908(n), with equality if and only if n is an exponentially squarefree number (A209061).

a(n) >= 1, with equality if and only if n is an exponentially powerful number (A361177).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.48739698475309778834..., where f(x) = (1-x) * (1 + Sum_{k>=1} A056671(k) * x^k).

Mathematica

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

Prog

(PARI) s(e) = vecprod(apply(x -> if(x == 1, 2, 1), factor(e)[, 2]));
a(n) = vecprod(apply(s, factor(n)[, 2]));

Crossrefs

Cf. A056624, A056671, A209061, A278908, A361177, A361255, A366309, A384557.

Sequence in context: A194799 A291119 A366309 * A394425 A388973 A391988

Adjacent sequences: A392447 A392448 A392449 * A392451 A392452 A392453

Keyword

nonn,mult,easy

Author

Amiram Eldar, Jan 13 2026

Status

approved