A391427 Cubefree exponential abundant numbers: cubefree numbers k for which A051377(k) > 2*k.

900, 1764, 4356, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 49284, 51156, 52020, 53100

Offset

1,1

Comments

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

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

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.

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)).

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

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.

Links

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

Mathematica

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]

Prog

(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);

Crossrefs

Intersection of A004709 and A129575.

Subsequence of A357695.

Subsequences: A391428, A391429, A391430.

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

Sequence in context: A383694 A383698 A386798 * A074853 A392741 A391320

Adjacent sequences: A391424 A391425 A391426 * A391428 A391429 A391430

Keyword

nonn,easy

Author

Amiram Eldar, Dec 09 2025

Status

approved