A348963 - OEIS

%I #10 Aug 06 2024 06:06:34

%S 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,

%T 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,

%U 1,1,3,1,1,1,20,1,1,6,3,1,1,1,13,37,1,1,3

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

%C The function S_e(n) in Sándor (2006).

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

%H Amiram Eldar, <a href="/A348963/b348963.txt">Table of n, a(n) for n = 1..10000</a>

%H József Sándor, <a href="http://citeseerx.ist.psu.edu/pdf/2936ca1cfcb9e3673ed4165dca32dbee1f4070f5">On exponentially harmonic numbers</a>, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.

%H József Sándor, <a href="https://blngcc.files.wordpress.com/2008/11/jozsel-sandor-selected-chaters-of-geometry-analysis-and-number-theory.pdf">Selected Chapters of Geomety, Analysis and Number Theory</a>, 2005, pp. 141-145.

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

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

%Y Cf. A005117, A049419, A348964.

%K nonn,mult

%O 1,4

%A _Amiram Eldar_, Nov 05 2021