A349025 - OEIS

%I #10 Nov 06 2021 12:04:55

%S 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,

%T 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,

%U 1,3,1,1,1,20,1,1,6,3,1,1,1,9,28,1,1,3,1

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

%C First differs from A348963 at n = 16.

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

%H Amiram Eldar, <a href="/A349025/b349025.txt">Table of n, a(n) for n = 1..10000</a>

%H Nicuşor Minculete, <a href="http://www.imar.ro/~purice/Inst/2012/Minculete-Dr.pdf">Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor</a>, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.

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

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

%Y The unitary version of A348963.

%Y Cf. A005117, A278908, A349026.

%K nonn,mult

%O 1,4

%A _Amiram Eldar_, Nov 06 2021