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A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2*d = 2*k, where esigma is the sum of exponential divisors function (A051377).

%I A336680 #9 Jul 31 2020 06:46:38
%S A336680 900,1764,4356,4500,4900,6084,6300,7056,8820,9900,10404,11700,12348,
%T A336680 12996,14700,15300,17100,19044,19404,20700,21780,22932,26100,27900,
%U A336680 29988,30276,30420,30492,31500,33300,33516,34596,35280,36900,38700,40572,42300,42588,47700
%N A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2*d = 2*k, where esigma is the sum of exponential divisors function (A051377).
%C A336680 Equivalently, numbers that are equal to the sum of their proper exponential divisors, with one of them taken with a minus sign.
%H A336680 Amiram Eldar, <a href="/A336680/b336680.txt">Table of n, a(n) for n = 1..10000</a>
%e A336680 900 is a term since 900 = 30 + 60 + 90 + 150 - 180 + 300 + 450 is the sum of its proper exponential divisors with one of them, 180, taken with a minus sign.
%t A336680 dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; expAdmQ[n_] := (ab = esigma[n] - 2*n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && expDivQ[n, ab/2]; Select[Range[50000], expAdmQ]
%Y A336680 The exponential version of A111592.
%Y A336680 Subsequence of A129575.
%Y A336680 Similar sequences: A328328, A334972, A334974.
%Y A336680 Cf. A051377, A322791.
%K A336680 nonn
%O A336680 1,1
%A A336680 _Amiram Eldar_, Jul 30 2020