A087248 Squarefree abundant numbers.

30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 210, 222, 246, 258, 282, 318, 330, 354, 366, 390, 402, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 642, 654, 678, 690, 714, 762, 770, 786, 798, 822, 834, 858, 870, 894, 906, 910, 930, 942, 966, 978

Offset: 1

Author

Labos Elemer, Sep 05 2003

Keywords

• nonn

Comments

First odd term is 15015 = 3 * 5 * 7 * 11 * 13, with 32 divisors that add up to 32256 = 2*15015 + 2226. See A112643. - Alonso del Arte, Nov 06 2017

The lower asymptotic density of this sequence is larger than 1/(2*Pi^2) = 0.05066... which is the density of its subsequence of squarefree numbers larger than 6 and divisible by 6. The number of terms below 10^k for k=1,2,... is 0, 5, 53, 556, 5505, 55345, 551577, 5521257, 55233676, 552179958, 5521420147, ..., so it seems that this sequence has an asymptotic density which equals to about 0.05521... - Amiram Eldar, Feb 13 2021

The asymptotic density of this sequence is larger than 0.0544 (Wall, 1970). - Amiram Eldar, Apr 18 2024

Examples

			Checking that 30 = 2 * 3 * 5 and sigma(30) = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72, which is more than twice 30, we verify that 30 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Crossrefs

Cf. A087244, A087245, A087246, A087247, A087248, A008683, A005117, A005101, A112643.

Programs

Maple

isA005101 := proc(n)
    simplify(numtheory[sigma](n)>2*n);
end proc:
isA087248 := proc(n)
    isA005101(n) and numtheory[issqrfree](n) ;
end proc:
for n from 1 to 500 do
    if isA087248(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

Mathematica

Select[Range[10^3], SquareFreeQ@ # && DivisorSigma[1, #] > 2 # &] (* Michael De Vlieger, Feb 05 2017 *)

PARI

isA087248(i) = (sigma(i) > 2*i) && issquarefree(i) \\ Michel Marcus, Mar 09 2013

Formula

A005117 INTERSECT A005101.