A381740 Squarefree numbers k such that k^2 is abundant.

6, 10, 14, 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 170, 174, 182, 186, 190, 210, 222, 230, 238, 246, 258, 266, 282, 286, 290, 310, 318, 322, 330, 354, 366, 370, 374, 390, 402, 406, 410, 418, 426, 430, 434, 438, 442, 462, 470, 474, 498, 506, 510

Offset: 1

Author

Amiram Eldar, Mar 05 2025

Keywords

• nonn
• easy

Comments

The least odd term is a(105) = 105.

The least term that is coprime to 6 is a(3637276) = 37182145.

If k is a squarefree number that is divisible by 6, 10 or 14, then it is a term. Therefore a lower bound for the asymptotic density of this sequence is 29/(192*zeta(2)) = 0.0918... .

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 8, 99, 972, 9826, 97610, 979190, 9770801, 97650638, 976893969, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0976... .

If k is a term then any multiple of k that is squarefree is a term. The primitive terms are in A381741.

Links

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

Crossrefs

Intersection of A005117 and A381738.

Subsequences: A087248, A381741.

Cf. A013661.

Programs

Mathematica

Select[Range[600], SquareFreeQ[#] && DivisorSigma[-1, #^2] > 2 &]

PARI

isok(k) = {my(f = factor(k)); if(!issquarefree(f), 0, prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2);}