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

1, 2, 6, 30, 42, 66, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1410, 1518, 1554, 1590, 1722, 1770, 1806, 1914, 1974, 2046, 2226, 2310, 2442, 2478, 2562, 2706, 2730, 2814, 2838, 2982, 3066, 3102, 3318, 3486, 3498

Offset: 1

Author

Frank M Jackson, Jan 21 2025

Keywords

• nonn

Comments

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.

Similar to and a subsequence of A265501.

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

Examples

			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.
		

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000
• Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
• Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.

Crossrefs

Intersection of A005117 and A174973.

Subsequence of A267124 and of A265501.

Cf. A005153.

Programs

Maple

filter:= proc(n) local D,i;
  if not numtheory:-issqrfree(n) then return false fi;
  D:= sort(convert(numtheory:-divisors(n),list));
  andmap(i -> D[i+1]<=2*D[i],[$1..nops(D)-1])
end proc:
select(filter, [1,seq(i,i=2..5000,4)]); # Robert Israel, Jan 23 2025

Mathematica

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[#]&]

Formula

a(n) = C*n*log(n*log(n)) + O(n), where C = 14.56... (see comments). - Andreas Weingartner, Jan 23 2025