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

1,1

Comments

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

Links

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

Example

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.

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]

Crossrefs

The exponential version of A111592.

Subsequence of A129575.

Similar sequences: A328328, A334972, A334974.

Cf. A051377, A322791.

Sequence in context: A328136 A336254 A321206 * A074853 A162143 A258888

Adjacent sequences: A336677 A336678 A336679 * A336681 A336682 A336683

Keyword

nonn

Author

Amiram Eldar, Jul 30 2020

Status

approved