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

Author

Amiram Eldar, Oct 04 2019

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Intersection of A001694 and A129575.

Cf. A051377, A054979, A054980, A126164.

Programs

Mathematica

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