

A328136


Primitive exponential abundant numbers: the powerful terms of A129575.


1



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
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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. 343349.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465471.


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: A318720 A137490 A129575 * A321206 A074853 A162143
Adjacent sequences: A328133 A328134 A328135 * A328137 A328138 A328139


KEYWORD

nonn


AUTHOR

Amiram Eldar, Oct 04 2019


STATUS

approved



