A383694 Primitive exponential unitary abundant numbers: the powerful terms of A383693.

900, 1764, 4356, 4500, 4900, 6084, 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, 213444, 217800, 220500

Offset: 1

Author

Amiram Eldar, May 05 2025

Keywords

• nonn

Comments

First differs from its subsequence A383698 at n = 11.

For squarefree numbers k, eusigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857). Thus, if m is a term (eusigma(m) > 2*m) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) > 2*k*m, so k*m is an exponential unitary abundant number. Therefore, the sequence of exponential unitary abundant numbers (A383693) can be generated from this sequence by multiplying with coprime squarefree numbers.

The least odd term is a(1455) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.

Examples

			900 is a term since eusigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is a powerful number.
6300 is exponential unitary abundant, since eusigma(6300) = 15120 > 2 * 6300, but it is not a powerful number: 6300 = 2^2 * 3^2 * 5^2 * 7. Thus it is not in this sequence. It can be generated as a term of A383693 from a(1) = 900 by 7 * 900 = 6300, since 7 is squarefree and gcd(7, 900) = 1.
		

Links

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

Crossrefs

Intersection of A001694 and A383693.

A383698 is a subsequence.

Cf. A005117, A322857, A328136.

Programs

Mathematica

fun[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]

PARI

fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}