A391427 - OEIS

%I #10 Dec 10 2025 08:39:08

%S 900,1764,4356,4900,6084,6300,8820,9900,10404,11700,12996,14700,15300,

%T 17100,19044,19404,20700,21780,22932,26100,27900,29988,30276,30420,

%U 30492,33300,33516,34596,36900,38700,40572,42300,42588,44100,47700,49284,51156,52020,53100

%N Cubefree exponential abundant numbers: cubefree numbers k for which A051377(k) > 2*k.

%C Differs from A386798 by having the terms 44100, 108900, 152100, 213444, ..., and not having the terms 11025, 12100, 16900, 22050, ... .

%C If k and m are coprime terms, then k*m is also a term.

%C In particular, if k is a term, and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A391428) are the powerful (A001694) terms.

%C The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=1} 1/A064987(A087248(n)) = Sum_{n>=1} f(A391428(n)) = 0.000761256505..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)).

%C The least odd term is a(171626) = A391429(1) = 225450225 = (3 * 5 * 7 * 11 * 13)^2.

%C The least term that is coprime to 6 is a(8.505...*10^17) = 1117347505588495206025 = (5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31)^2.

%H Amiram Eldar, <a href="/A391427/b391427.txt">Table of n, a(n) for n = 1..10000</a>

%t f[p_, e_] := DivisorSum[e, p^# &]; q[1] = False; q[n_] := Module[{fct = FactorInteger[n]}, AllTrue[fct[[;;, 2]], # < 3 &] && Times @@ f @@@ fct > 2*n]; Select[Range[55000], q]

%o (PARI) isok(k) = if(k < 2, 0, my(f = factor(k)); vecmax(f[ ,2]) < 3 && prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) > 2*k);

%Y Intersection of A004709 and A129575.

%Y Subsequence of A357695.

%Y Subsequences: A391428, A391429, A391430.

%Y Cf. A001694, A005117, A051377, A059956, A064987, A087248, A386798.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Dec 09 2025