A328136 Primitive exponential abundant numbers: the powerful terms of A129575.

900, 1764, 3600, 4356, 4500, 4900, 6084, 7056, 8100, 10404, 12348, 12996, 19044, 22500, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 86436, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176400, 176868, 181476, 191844

Offset

1,1

Comments

For squarefree numbers k, esigma(k) = k, where esigma is the sum of exponential divisors function (A051377). Thus, if m is a term (esigma(m) > 2m) and k is a squarefree number coprime to m, then esigma(k*m) = esigma(k) * esigma(m) = k * esigma(m) > 2*k*m, so k*m is an exponential abundant number. Therefore the sequence of exponential abundant numbers (A129575) can be generated from this sequence by multiplying with coprime squarefree numbers.

Links

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

Peter Hagis, Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2 (1988), pp. 343-349.

E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471.

Example

900 is a term since esigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is powerful.
6300 is exponential abundant, since esigma(6300) = 15120 > 2 * 6300, but it is not powerful, 6300 = 2^2 * 3^2 * 5^2 * 7, thus it is not in this sequence. It can be generated as a term of A129575 from 900 by 7 * 900 = 6300, since gcd(7, 900) = 1.

Mathematica

fun[p_, e_] := DivisorSum[e, p^# &]; aQ[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2n; Select[Range[200000], aQ]

Crossrefs

Intersection of A001694 and A129575.

Cf. A051377, A054979, A054980, A126164.

Sequence in context: A338540 A137490 A129575 * A336254 A321206 A336680

Adjacent sequences: A328133 A328134 A328135 * A328137 A328138 A328139

Keyword

nonn

Author

Amiram Eldar, Oct 04 2019

Status

approved