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A196149
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Numbers whose divisors increase by a factor of 3 or less.
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4
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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, 50, 54, 56, 60, 63, 64, 66, 70, 72, 75, 78, 80, 81, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 117, 120, 126, 128, 130, 132, 135, 136, 140, 144, 147, 150
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OFFSET
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1,2
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COMMENTS
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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
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
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LINKS
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FORMULA
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a(n) = C*n*log(n*log(n)) + O(n), where C = 0.486513… (See comments). - Andreas Weingartner, Jun 25 2021
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EXAMPLE
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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
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MATHEMATICA
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dif3[n_]:=Max[#[[2]]/#[[1]]&/@Partition[Divisors[n], 2, 1]]<=3; Select[ Range[ 200], dif3] (* Harvey P. Dale, Jun 08 2015 *)
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PROG
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(Haskell)
a196149 n = a196149_list !! (n-1)
a196149_list = filter f [1..] where
f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 3) divs)
where divs = a027750_row' n
(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
(Python)
from sympy import divisors
def ok(n):
d = divisors(n)
return all(d[i]/d[i-1] <= 3 for i in range(1, len(d)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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