OFFSET
1,2
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
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.
FORMULA
a(n) = C*n*log(n*log(n)) + O(n), where C = 14.56... (see comments). - Andreas Weingartner, Jan 23 2025
EXAMPLE
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.
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[#]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jan 21 2025
STATUS
approved