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A087248
Squarefree abundant numbers.
12
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
1,1
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
LINKS
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.
FORMULA
A005117 INTERSECT A005101.
EXAMPLE
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.
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 *)
PROG
(PARI) isA087248(i) = (sigma(i) > 2*i) && issquarefree(i) \\ Michel Marcus, Mar 09 2013
KEYWORD
nonn
AUTHOR
Labos Elemer, Sep 05 2003
STATUS
approved