A349025 a(n) is multiplicative with a(p^e) = Sum_{d||e} p^(e-d), where d||e are the unitary divisors of e.

1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 9, 1, 4, 1, 3, 1, 1, 1, 5, 6, 1, 10, 3, 1, 1, 1, 17, 1, 1, 1, 12, 1, 1, 1, 5, 1, 1, 1, 3, 4, 1, 1, 9, 8, 6, 1, 3, 1, 10, 1, 5, 1, 1, 1, 3, 1, 1, 4, 57, 1, 1, 1, 3, 1, 1, 1, 20, 1, 1, 6, 3, 1, 1, 1, 9, 28, 1, 1, 3, 1

Offset: 1

Author

Amiram Eldar, Nov 06 2021

Keywords

• nonn
• mult

Comments

First differs from A348963 at n = 16.

A number k is an exponential unitary harmonic number (A349026) if and only if a(k) | k * A278908(k).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Nicuşor Minculete, Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.

Crossrefs

The unitary version of A348963.

Cf. A005117, A278908, A349026.

Programs

Mathematica

f[p_, e_] := DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Formula

a(n) = 1 if and only if n is squarefree (A005117).