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%I #30 Jun 26 2021 02:31:30
%S 1,2,3,4,6,8,9,10,12,15,16,18,20,21,24,27,28,30,32,36,40,42,44,45,48,
%T 50,54,56,60,63,64,66,70,72,75,78,80,81,84,88,90,96,99,100,102,104,
%U 105,108,110,112,117,120,126,128,130,132,135,136,140,144,147,150
%N Numbers whose divisors increase by a factor of 3 or less.
%C The polymath8 project led by Terry Tao refers to these numbers as "3-densely divisible". In general they say that n is y-densely divisible if its divisors increase by a factor of y or less, or equivalently, if for every R with 1 <= R <= n, there is a divisor in the interval [R/y,R]. They use this as a weakening of the condition that n be y-smooth. - _David S. Metzler_, Jul 02 2013
%C Let D(x) denote the number of such integers up to x. D(x) has order of magnitude x/log(x) (See Saias 1997). Moreover, we have D(x) = c*x/log(x) + O(x/(log(x))^2), where c = 2.05544... (See Weingartner 2015, 2019). As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 0.486513... - _Andreas Weingartner_, Jun 25 2021
%H Reinhard Zumkeller, <a href="/A196149/b196149.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 Terence Tao, <a href="http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/">A Truncated Elementary Selberg Sieve of Pintz</a>. (blog entry defining y-densely divisible)
%H Terence Tao et al., <a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes">Polymath8 home page</a>.
%H Andreas Weingartner, <a href="https://doi.org/10.1093/qmath/hav006">Practical numbers and the distribution of divisors</a>, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; <a href="https://arxiv.org/abs/1405.2585">arXiv preprint</a>, arXiv:1405.2585 [math.NT], 2014-2015.
%H 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 a(n) = A052287(n) / 3.
%F a(n) = C*n*log(n*log(n)) + O(n), where C = 0.486513… (See comments). - _Andreas Weingartner_, Jun 25 2021
%e 14 is not a term because its divisors are 1,2,7,14, and the gap from 2 to 7 is more than a factor of 3. - _N. J. A. Sloane_, Aug 03 2015
%t dif3[n_]:=Max[#[[2]]/#[[1]]&/@Partition[Divisors[n],2,1]]<=3; Select[ Range[ 200],dif3] (* _Harvey P. Dale_, Jun 08 2015 *)
%o (Haskell)
%o a196149 n = a196149_list !! (n-1)
%o a196149_list = filter f [1..] where
%o f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 3) divs)
%o where divs = a027750_row' n
%o -- _Reinhard Zumkeller_, Jun 25 2015, Sep 28 2011
%o (PARI) is(n)=my(d=divisors(n));for(i=2,#d,if(d[i]>3*d[i-1],return(0)));1 \\ _Charles R Greathouse IV_, Jul 06 2013
%o (Python)
%o from sympy import divisors
%o def ok(n):
%o d = divisors(n)
%o return all(d[i]/d[i-1] <= 3 for i in range(1, len(d)))
%o print(list(filter(ok, range(1, 151)))) # _Michael S. Branicky_, Jun 25 2021
%Y A174973 is a subsequence.
%Y Cf. A027750.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Sep 28 2011
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