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A380328 2-dense squarefree numbers: Squarefree numbers whose divisors increase by factors of at most 2.

%I A380328 #28 Jan 25 2025 12:33:42
%S A380328 1,2,6,30,42,66,210,330,390,462,510,546,570,690,714,798,858,870,930,
%T A380328 966,1110,1122,1218,1230,1254,1290,1302,1410,1518,1554,1590,1722,1770,
%U A380328 1806,1914,1974,2046,2226,2310,2442,2478,2562,2706,2730,2814,2838,2982,3066,3102,3318,3486,3498
%N A380328 2-dense squarefree numbers: Squarefree numbers whose divisors increase by factors of at most 2.
%C A380328 This sequence is a subsequence of primitive practical numbers (A267124) because the sequence of 2-dense numbers (A174973) is a subsequence of practical numbers (A005153) and all squarefree practical numbers (A265501) are by definition primitive practical numbers.
%C A380328 Similar to and a subsequence of A265501.
%C A380328 Let N(x) be the number of terms less than x. Saias (1997) showed that N(x) has order of magnitude x/log(x). We have N(x) = c*x/log(x) + O(x/(log(x))^2), where c=0.06864... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 14.56... (see Weingartner (2019)). - _Andreas Weingartner_, Jan 23 2025
%H A380328 Frank M Jackson, <a href="/A380328/b380328.txt">Table of n, a(n) for n = 1..10000</a>
%H A380328 Eric Saias, <a href="https://doi.org/10.1006/jnth.1997.2057">Entiers à diviseurs denses 1</a>, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
%H A380328 Andreas Weingartner, <a href="https://doi.org/10.1090/mcom/3402">On the constant factor in several related asymptotic estimates</a>, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; <a href="https://arxiv.org/abs/1705.06349">arXiv preprint</a>, arXiv:1705.06349 [math.NT], 2017-2018.
%F A380328 a(n) = C*n*log(n*log(n)) + O(n), where C = 14.56... (see comments). - _Andreas Weingartner_, Jan 23 2025
%e A380328 a(5) = 42 and its prime factorization is 2*3*7 and squarefree. Also the proper divisors are 1, 2, 3, 6, 7, 21, 42 they are 2-dense and therefore 42 is practical as well as being primitive practical.
%p A380328 filter:= proc(n) local D,i;
%p A380328   if not numtheory:-issqrfree(n) then return false fi;
%p A380328   D:= sort(convert(numtheory:-divisors(n),list));
%p A380328   andmap(i -> D[i+1]<=2*D[i],[$1..nops(D)-1])
%p A380328 end proc:
%p A380328 select(filter, [1,seq(i,i=2..5000,4)]); # _Robert Israel_, Jan 23 2025
%t A380328 Dens2DivQ[n_] := Module[{lst=Divisors[n], m, ok}, If[n==1, Return[True]]; Do[ok=False; If[lst[[m+1]]/lst[[m]]>2, Break[]]; ok=True, {m, 1, Length[lst]-1}]; ok]; Select[Range[10000], SquareFreeQ[#]&&Dens2DivQ[#]&]
%Y A380328 Intersection of A005117 and A174973.
%Y A380328 Subsequence of A267124 and of A265501.
%Y A380328 Cf. A005153.
%K A380328 nonn
%O A380328 1,2
%A A380328 _Frank M Jackson_, Jan 21 2025