A348964 - OEIS

%I #11 Aug 06 2024 06:06:02

%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 harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

%C Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 2.

%C Equivalently, numbers k such that A348963(k) | k * A049419(k).

%C Apparently, most exponential harmonic numbers of type 1 (A348961) are also terms of this sequence. Those that are not exponential harmonic numbers of type 2 are 1936, 5808, 9680, 13552, 17424, 29040, ...

%H Amiram Eldar, <a href="/A348964/b348964.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.

%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.

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

%e 12 is a term since its exponential divisors are 6 and 12, and their harmonic mean, 8, is an integer.

%t f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^(e-#) &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

%Y A005117 and A348965 are subsequences.

%Y Cf. A049419, A322791, A348961, A348963.

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

%K nonn

%O 1,2

%A _Amiram Eldar_, Nov 05 2021