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A112643 Odd squarefree abundant numbers.

%I A112643 #30 Jan 16 2025 02:38:13
%S A112643 15015,19635,21945,23205,25935,26565,31395,33495,33915,35805,39585,
%T A112643 41055,42315,42735,45885,47355,49665,50505,51765,54285,55965,58695,
%U A112643 61215,64155,68145,70455,72345,77385,80535,82005,83265,84315,91245
%N A112643 Odd squarefree abundant numbers.
%C A112643 Deviates from A046391 (does not contain 36465, 40755 for example).
%C A112643 The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 134, 1663, 16328, 175630, 1694621, 16726454, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00016... . - _Amiram Eldar_, Sep 02 2022
%C A112643 From _Amiram Eldar_, Jan 15 2025: (Start)
%C A112643 The least term that is not divisible by 5 is a(3696) = 22309287.
%C A112643 The least term that is not divisible by 3 is a(5607800) = 33426748355.
%C A112643 The least term that is coprime to 15 is 1357656019974967471687377449. (End)
%H A112643 Donovan Johnson, <a href="/A112643/b112643.txt">Table of n, a(n) for n = 1..10000</a>
%F A112643 A087248 INTERSECT A005408.
%F A112643 omega(a(n)) >= 5, where omega(n) = A001221(n) is the number of distinct primes dividing n. - _Amiram Eldar_, Jan 15 2025
%e A112643 199815 = 3 * 5 * 7 * 11 * 173, with 32 divisors adding up to 400896 = 2 * 199815 + 1266.
%p A112643 # see A087248 for the additional code
%p A112643 isA112643 := proc(n)
%p A112643     isA087248(n) and type(n,'odd') ;
%p A112643 end proc:
%p A112643 for n from 1 do
%p A112643     if isA112643(n) then
%p A112643         print(n);
%p A112643     end if;
%p A112643 end do: # _R. J. Mathar_, Nov 10 2014
%t A112643 ta = {{0}}; Do[g = n; s = DivisorSigma[1, n] - 2 * n; If[Greater[s, 0] && Equal[Abs[MoebiusMu[n]], 1] && !Equal[Mod[n, 2], 0], Print[n, PrimeFactorList[n], s]; ta = Append[ta, n]], {n, 1, 200000}];{ta = Delete[ta, 1], g}(* Elemer *)
%t A112643 Select[Range[1, 99999, 2], MoebiusMu[#] != 0 && DivisorSigma[1, #] > 2 # &] (* _Alonso del Arte_, Nov 11 2017 *)
%o A112643 (PARI) is(n)=if(n%2==0, return(0)); my(f=factor(n)); sigma(f)>2*n && vecmax(f[,2])==1 \\ _Charles R Greathouse IV_, Feb 21 2017
%Y A112643 Cf. A001221, A005101, A005231, A005408, A046391, A087248.
%K A112643 nonn
%O A112643 1,1
%A A112643 _Labos Elemer_, Sep 20 2005