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A349026 Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.

%I A349026 #15 Nov 06 2021 12:04:53
%S A349026 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 A349026 36,37,38,39,40,41,42,43,46,47,51,53,55,57,58,59,60,61,62,65,66,67,69,
%U A349026 70,71,73,74,75,77,78,79,82,83,84,85,86,87,89,90,91,93,94
%N A349026 Exponential unitary harmonic numbers: numbers k such that the harmonic mean of the exponential unitary divisors of k is an integer.
%C A349026 First differs from A348964 at n = 102. a(102) = 144 is not an exponential harmonic number of type 2.
%C A349026 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 A349026 Equivalently, numbers k such that A349025(k) | k * A278908(k).
%H A349026 Amiram Eldar, <a href="/A349026/b349026.txt">Table of n, a(n) for n = 1..10000</a>
%H A349026 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 A349026 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 A349026 144 is a term since its exponential unitary divisors are 6, 18, 48 and 144, and their harmonic mean, 16, is an integer.
%t A349026 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 A349026 Cf. A278908 (number of exponential unitary divisors), A322857, A322858, A323310, A349025, A349027.
%Y A349026 Similar sequences: A001599, A006086, A063947, A286325, A319745, A348964.
%K A349026 nonn
%O A349026 1,2
%A A349026 _Amiram Eldar_, Nov 06 2021