A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500

Offset: 1

Author

Amiram Eldar, May 05 2025

Keywords

• nonn
• easy

Comments

First differs from its subsequence A383697 at n = 21.

All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).

If an exponential abundant number (A129575) is exponentially squarefree (A209061), then it is in this sequence. Terms of this sequence that are not exponentially squarefree are a(21) = 22500, a(77) = 86436, a(140) = 157500, etc..

The least odd term is a(202273) = 225450225, and the least term that is coprime to 6 is a(1.002..*10^18) = 1117347505588495206025.

The asymptotic density of this sequence is Sum_{n>=1} f(A383694(n)) = 0.00089722..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

Examples

			900 is a term since A322857(900) = 2160 > 2*900 = 1800.
		

Links

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

Crossrefs

Cf. A005117, A209061, A322857, A322858, A361255.

Subsequence of A013929 and A129575.

Subsequences: A383694, A383697, A383698.

Programs

Mathematica

f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]

PARI

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