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Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).
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%I #9 Jan 18 2020 03:12:57

%S 1,2,3,4,7,8,9,12,16,18,20,28,36,52,60,68,72,84,92,100,124,132,140,

%T 144,180,244,252,300,324,360,396,468,588,612,684,828,900,1116,1260,

%U 1332,1476,1548,1692,1764,2124,2196,2340,2412,2556,2628,2700,2772,2844,2988

%N Exponential highly abundant numbers: numbers m such that esigma(m) > esigma(k) for all k < m, where esigma(m) is the sum of exponential divisors of m (A051377).

%C The exponential version of A002093.

%H Amiram Eldar, <a href="/A328134/b328134.txt">Table of n, a(n) for n = 1..1000</a>

%e The first 10 values of esigma(k) for k = 1 to 10 are 1, 2, 3, 6, 5, 6, 7, 10, 12, 10. The record values are reached for 1, 2, 3, 4, 7, 8, 9.

%t f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; s = {}; em = 0; Do[e = esigma[n]; If[e > em, em = e; AppendTo[s, n]], {n, 1, 3000}]; s

%Y Cf. A002093, A051377, A285614 (unitary), A292983 (bi-unitary), A327634 (infinitary).

%K nonn

%O 1,2

%A _Amiram Eldar_, Oct 04 2019