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A174973 Numbers whose divisors increase by a factor of 2 or less. 23
1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

That is, if the divisors of a number are listed in increasing order, the ratio of adjacent divisors is at most 2. The only odd number in this sequence is 1. Every term appears to be a practical number (A005153). The first practical number not here is 78.

Let p1^e1*p2^e2...pr^er be the prime factorization of a number, with primes p1 < p2 < ... < pr and ek > 0. Then the number is in this sequence if and only if pk <= 2*Product_{j < k} p_j^e_j. This condition is similar to, but more restrictive than, the condition for practical numbers.

The polymath8 project led by Terry Tao refers to these numbers as "2-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

Also, numbers n with the property that the symmetric representation of sigma(n) has only one part. See A238443 and A237593. - Omar E. Pol, Mar 06 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

T. Tao, A Truncated Elementary Selberg Sieve of Pintz (blog entry defining y-densely divisible)

T. Tao et al., Polymath8 home page

FORMULA

a(n) = A047836(n) / 2. - Reinhard Zumkeller, Sep 28 2011

EXAMPLE

The divisors of 12 are 1, 2, 3, 4, 6, 12. The ratios of adjacent divisors is 2, 3/2, 4/3, 3/2, and 2, which are all <=2. Hence 12 is in this sequence.

MAPLE

a:= proc() option remember; local k; for k from 1+`if`(n=1, 0,

      a(n-1)) while (l-> ormap(x-> x, [seq(l[i]>l[i-1]*2, i=2..

      nops(l))]))(sort([(numtheory[divisors](k))[]])) do od; k

    end:

seq(a(n), n=1..100);  # Alois P. Heinz, Jul 27 2018

MATHEMATICA

OK[n_] := Module[{d=Divisors[n]}, And@@(#<=2& /@ (Rest[d]/Most[d]))]; Select[Range[1000], OK]

PROG

(Haskell)

a174973 n = a174973_list !! (n-1)

a174973_list = filter f [1..] where

   f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 2) 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]>2*d[i-1], return(0))); 1 \\ Charles R Greathouse IV, Jul 06 2013

CROSSREFS

Subsequence of A196149. Same as A238443.

Cf. A027750, A237593.

Sequence in context: A125225 A092903 A005153 * A238443 A231565 A191743

Adjacent sequences:  A174970 A174971 A174972 * A174974 A174975 A174976

KEYWORD

nonn

AUTHOR

T. D. Noe, Apr 02 2010

EXTENSIONS

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

STATUS

approved

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Last modified April 22 11:46 EDT 2019. Contains 322330 sequences. (Running on oeis4.)