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).

900, 1764, 4356, 4500, 4900, 6084, 6300, 7056, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 40572, 42300, 42588, 47700

Offset: 1

Author

Amiram Eldar, Jul 30 2020

Keywords

• nonn

Comments

Equivalently, numbers that are equal to the sum of their proper exponential divisors, with one of them taken with a minus sign.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

The exponential version of A111592.

Subsequence of A129575.

Similar sequences: A328328, A334972, A334974.

Cf. A051377, A322791.

Programs

Mathematica

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]