cp's OEIS Frontend

A087248 Squarefree abundant numbers.

%I A087248 #40 Apr 18 2024 04:22:00
%S A087248 30,42,66,70,78,102,114,138,174,186,210,222,246,258,282,318,330,354,
%T A087248 366,390,402,426,438,462,474,498,510,534,546,570,582,606,618,642,654,
%U A087248 678,690,714,762,770,786,798,822,834,858,870,894,906,910,930,942,966,978
%N A087248 Squarefree abundant numbers.
%C A087248 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
%C A087248 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
%C A087248 The asymptotic density of this sequence is larger than 0.0544 (Wall, 1970). - _Amiram Eldar_, Apr 18 2024
%H A087248 Robert Israel, <a href="/A087248/b087248.txt">Table of n, a(n) for n = 1..10000</a>
%H A087248 Charles Robert Wall, <a href="https://trace.tennessee.edu/utk_graddiss/8570/">Topics related to the sum of unitary divisors of an integer</a>, Ph.D. diss., University of Tennessee, 1970.
%F A087248 A005117 INTERSECT A005101.
%e A087248 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.
%p A087248 isA005101 := proc(n)
%p A087248     simplify(numtheory[sigma](n)>2*n);
%p A087248 end proc:
%p A087248 isA087248 := proc(n)
%p A087248     isA005101(n) and numtheory[issqrfree](n) ;
%p A087248 end proc:
%p A087248 for n from 1 to 500 do
%p A087248     if isA087248(n) then
%p A087248         print(n);
%p A087248     end if;
%p A087248 end do: # _R. J. Mathar_, Nov 10 2014
%t A087248 Select[Range[10^3], SquareFreeQ@ # && DivisorSigma[1, #] > 2 # &] (* _Michael De Vlieger_, Feb 05 2017 *)
%o A087248 (PARI) isA087248(i) = (sigma(i) > 2*i) && issquarefree(i) \\ _Michel Marcus_, Mar 09 2013
%Y A087248 Cf. A087244, A087245, A087246, A087247, A087248, A008683, A005117, A005101, A112643.
%K A087248 nonn
%O A087248 1,1
%A A087248 _Labos Elemer_, Sep 05 2003