A357695 Cubefree abundant numbers.

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 114, 126, 132, 138, 140, 150, 156, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 246, 252, 258, 260, 276, 282, 294, 300, 306, 308, 318, 330, 340, 342, 348, 350, 354, 364, 366, 372, 380, 390, 396

Offset: 1

Author

Amiram Eldar, Oct 10 2022

Keywords

• nonn

Comments

The least odd term is a(224) = A357697(1) = 1575.

The lower asymptotic density of this sequence is larger than 12/(91*zeta(3)) = 0.1097... which is the density of its subsequence of cubefree numbers larger than 6 and divisible by 6.

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 13, 143, 1440, 14470, 144187, 1442500, 14426015, 144267400, 1442567879, 14425142573, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1442... .

Examples

			12 = 2^2 * 3 is a term since it is cubefree and sigma(12) = 28 > 2*12.
		

Links

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

Crossrefs

Intersection of A004709 and A005101.

Subsequences: A087248, A357696, A357697.

Cf. A000203 (sigma), A002117, A308618.

Programs

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1); q[1] = False; q[n_] := AllTrue[(fct = FactorInteger[n])[[;;, 2]], # < 3 &] && Times @@ f @@@ fct > 2*n; Select[Range[400], q]

PARI

is(n) = {my(f = factor(n)); (n==1 || vecmax(f[,2]) < 3) && sigma(f, -1) > 2};