A348963 a(n) is multiplicative with a(p^e) = Sum_{d|e} p^(e-d).

1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 13, 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, 13, 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, 13, 37, 1, 1, 3

Offset: 1

Author

Amiram Eldar, Nov 05 2021

Keywords

• nonn
• mult

Comments

The function S_e(n) in Sándor (2006).

A number k is an exponential harmonic of type 2 (A348964) if and only if a(k) | k * A049419(k).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
• József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

Crossrefs

Cf. A005117, A049419, A348964.

Programs

Mathematica

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

Formula

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