login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A174973 Numbers whose divisors increase by a factor of 2 or less. 35
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

Saias (1997) called these terms "2-dense numbers" and proved that if N(x) is the number of terms not exceeding x, then there are two positive constants c_1 and c_2 such that c_1 * x/log_2(x) <= N(x) <= c_2 * x/log_2(x) for all x >= 2. - Amiram Eldar, Jul 23 2020

Weingartner (2015, 2019) showed that N(x) = c*x/log(x) + O(x/(log(x))^2), where c=1.224830... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 0.816439... - Andreas Weingartner, Jun 22 2021

Subsequence of A071562. All powers of 2 (cf. A000079) and all even perfect numbers (cf. A000396) are in the sequence. - Omar E. Pol, Aug 04 2021

LINKS

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

José Manuel Rodríguez Caballero, Jordan's Expansion of the Reciprocal of Theta Functions and 2-densely Divisible Numbers, Integers, Vol. 20 (2020), Article A2.

Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.

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

Terence Tao et al., Polymath8 home page.

Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015.

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

a(n) = C*n*log(n*log(n)) + O(n), where C = 0.816439... (See comments). - Andreas Weingartner, Jun 23 2021

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]

dif2Q[n_]:=AllTrue[#[[2]]/#[[1]]&/@Partition[Divisors[n], 2, 1], #<=2&]; Select[Range[300], dif2Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 29 2020 *)

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

(MAGMA) [k:k in [1..260]|forall{i:i in [1..#Divisors(k)-1]|d[i+1]/d[i] le 2 where d is Divisors(k)}]; // Marius A. Burtea, Jan 09 2020

(Python)

from sympy import divisors

def ok(n):

    d = divisors(n)

    return all(d[i]/d[i-1] <= 2 for i in range(1, len(d)))

print(list(filter(ok, range(1, 257)))) # Michael S. Branicky, Jun 22 2021

CROSSREFS

Subsequence of A196149. Same as A238443.

Cf. A027750, A047836, A237593.

Sequence in context: A125225 A092903 A005153 * A238443 A325795 A325781

Adjacent sequences:  A174970 A174971 A174972 * A174974 A174975 A174976

KEYWORD

nonn

AUTHOR

T. D. Noe, Apr 02 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 17 17:47 EST 2022. Contains 350402 sequences. (Running on oeis4.)