%I #15 Nov 06 2021 12:04:53

%S 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,21,22,23,26,29,30,31,33,34,35,

%T 36,37,38,39,40,41,42,43,46,47,51,53,55,57,58,59,60,61,62,65,66,67,69,

%U 70,71,73,74,75,77,78,79,82,83,84,85,86,87,89,90,91,93,94

%N Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.

%C First differs from A348964 at n = 102. a(102) = 144 is not an exponential harmonic number of type 2.

%C The exponential unitary divisors of n = Product p(i)^e(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of e(i) (see A278908).

%C Equivalently, numbers k such that A349025(k) | k * A278908(k).

%H Amiram Eldar, <a href="/A349026/b349026.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.

%e The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential unitary divisor, k itself, and thus the harmonic mean of its exponential unitary divisors is also k, which is an integer.

%e 144 is a term since its exponential unitary divisors are 6, 18, 48 and 144, and their harmonic mean, 16, is an integer.

%t f[p_, e_] := p^e * 2^PrimeNu[e] / DivisorSum[e, p^(e - #) &, CoprimeQ[#, e/#] &]; euhQ[1] = True; euhQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], euhQ]

%Y Cf. A278908 (number of exponential unitary divisors), A322857, A322858, A323310, A349025, A349027.

%Y Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.

%K nonn

%O 1,2

%A _Amiram Eldar_, Nov 06 2021