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

1,4

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.

Formula

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

Mathematica

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

Crossrefs

The unitary version of A348963.

Cf. A005117, A278908, A349026.

Sequence in context: A369180 A351565 A254101 * A348963 A350470 A277604

Adjacent sequences: A349022 A349023 A349024 * A349026 A349027 A349028

Keyword

nonn,mult

Author

Amiram Eldar, Nov 06 2021

Status

approved