A348961 - OEIS

%I #13 Aug 06 2024 06:07:50

%S 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,36,37,

%T 38,39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,

%U 77,78,79,82,83,85,86,87,89,91,93,94,95,97,101,102,103,105

%N Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

%C First differs from A005117 at n = 24, from A333634 and A348499 at n = 47, and from A336223 at n = 63.

%C Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 1, and that an e-perfect number (A054979) is a term of this sequence if and only if at least one of the exponents in its prime factorization is not a perfect square.

%C Since all the e-perfect numbers are products of a primitive e-perfect number (A054980) and a coprime squarefree number, and all the known primitive e-perfect numbers have a nonsquare exponent in their prime factorizations, there is no known e-perfect number that is not in this sequence.

%H Amiram Eldar, <a href="/A348961/b348961.txt">Table of n, a(n) for n = 1..10000</a>

%H József Sándor, <a href="https://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 3 is a term since esigma(3) = 3, 3 * d_e(3) = 3 * 1, so esigma(3) | 3 * d_e(3).

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

%Y A005117 and A348962 are subsequences.

%Y Cf. A049419, A051377, A322791, A333634, A336223, A348499, A348964.

%K nonn

%O 1,2

%A _Amiram Eldar_, Nov 05 2021