

A196149


Numbers whose divisors increase by a factor of 3 or less.


4



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

1,2


COMMENTS

The polymath8 project led by Terry Tao refers to these numbers as "3densely divisible". In general they say that n is ydensely 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 ysmooth.  David S. Metzler, Jul 02 2013


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Tao, A Truncated Elementary Selberg Sieve of Pintz (blog entry defining ydensely divisible)
T. Tao et al.,Polymath8 home page


FORMULA

a(n) = A052287(n) / 3.


EXAMPLE

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


MATHEMATICA

dif3[n_]:=Max[#[[2]]/#[[1]]&/@Partition[Divisors[n], 2, 1]]<=3; Select[ Range[ 200], dif3] (* Harvey P. Dale, Jun 08 2015 *)


PROG

(Haskell)
a196149 n = a196149_list !! (n1)
a196149_list = filter f [1..] where
f n = all (<= 0) $ zipWith () (tail divs) (map (* 3) divs)
where divs = a027750_row' n
 Reinhard Zumkeller, Jun 25 2015, Sep 28 2011
(PARI) is(n)=my(d=divisors(n)); for(i=2, #d, if(d[i]>3*d[i1], return(0))); 1 \\ Charles R Greathouse IV, Jul 06 2013


CROSSREFS

A174973 is a subsequence.
Cf. A027750.
Sequence in context: A240911 A064150 A259227 * A240163 A269045 A067023
Adjacent sequences: A196146 A196147 A196148 * A196150 A196151 A196152


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Sep 28 2011


EXTENSIONS

Alternate nomenclature and links added by David S. Metzler, Jul 02 2013


STATUS

approved



